🧠:[<think>Okay, so I have this problem here: I need to find the monotonic decreasing interval of the function y equals lg x squared. The options are A through D, with C being (-∞, 0) and D being (0, +∞). Hmm, let me think. First, let's recall what lg means. I think in some contexts, lg is logarithm base 10, right? So lg x² would be log base 10 of x squared. So the function is y = log₁₀(x²). Alright, let's write that down: y = log₁₀(x²). Now, the question is about the monotonic decreasing intervals. Monotonic decreasing means that as x increases, y decreases. So I need to find where the function is decreasing. To do that, I should probably find the derivative of the function and see where it's negative, right? Because if the derivative is negative, the function is decreasing there. But before jumping into calculus, maybe I can simplify the function. Let me see. x squared is inside the logarithm. Remember that log(x²) can be rewritten using logarithm properties. Since log(a^b) = b log a, so log₁₀(x²) is 2 log₁₀|x|. Because x squared is always positive, but x could be positive or negative. However, logarithm of a negative number isn't defined, so x can't be zero. Therefore, the domain of the function is all real numbers except x = 0. So the function is y = 2 log|x|. That's interesting. So the function is actually y = 2 log|x|. So this function is defined for x > 0 and x < 0. So the graph of this function would be symmetric with respect to the y-axis because it's an even function, right? Because |x| is even, and log|x| would also be even. Then multiplied by 2, still even. So the function's graph should be symmetric across the y-axis. But the question is about monotonicity. Monotonic decreasing intervals. So let's consider the behavior for x > 0 and x < 0 separately. Because when x is positive, |x| is x, so y = 2 log x. When x is negative, |x| is -x, so y = 2 log(-x). Let's see the derivative for each case. First, for x > 0: y = 2 log x. The derivative dy/dx is 2 * (1/x) * (1/ln 10). Wait, hold on. The derivative of log base 10 of x is 1/(x ln 10). So derivative of 2 log x would be 2/(x ln 10). Since x is positive, the derivative here is positive. So on the interval (0, +∞), the function is increasing. Now, for x < 0: y = 2 log(-x). Then the derivative dy/dx is 2 * (1/(-x)) * (-1) * (1/ln 10). Wait, let me check that. Let me do it step by step. Let u = -x, so y = 2 log u. Then dy/dx = 2 * (1/u) * du/dx. Here, du/dx is -1. So dy/dx = 2 * (1/(-x)) * (-1) = 2/(x ln 10). Wait, no. Wait, hold on. Wait, log base 10 of u, derivative is 1/(u ln 10) * du/dx. So if u = -x, du/dx = -1. Therefore, dy/dx = 2 * [1/((-x) ln 10)] * (-1) = 2/(x ln 10). Wait, but x is negative here. So in this case, x is negative, so x is less than zero. So dy/dx is 2 divided by (x ln 10). Since x is negative, and ln 10 is positive, the derivative is negative. Therefore, on the interval (-∞, 0), the derivative is negative, so the function is decreasing there. So putting this together: the function y = lg x² is increasing on (0, +∞) and decreasing on (-∞, 0). Therefore, the monotonic decreasing interval is (-∞, 0). So the answer should be option C. Wait, but let me verify. Maybe I should also sketch the graph mentally. When x is positive, as x increases, log x² increases because x squared increases, so log x² increases. When x is negative, as x increases (approaching zero from the left), x becomes less negative, so x squared decreases, so log x² decreases. Therefore, the function is decreasing when x is negative and increasing when x is positive. So yes, the decreasing interval is (-∞, 0). So the answer is C. But wait, the options are given as B: (-∞,0), (0,+∞); C: (-∞,0); D: (0,+∞). The question is about the monotonic decreasing interval. So since the function is decreasing on (-∞,0), the answer is C. But wait, the function is composed of two separate intervals, right? Because the function is not defined at x=0. So technically, the function is decreasing on (-∞,0) and increasing on (0,+∞). So if the question is about the decreasing interval, it's just (-∞,0). Therefore, C is the correct answer. But let me check if there is any trick here. The original function is y = lg x². Sometimes, people might confuse this with (lg x) squared, but here it's log of x squared, so the square is inside the log. So that's different. So no, the function is correctly interpreted as log(x²). Alternatively, let's consider the function's behavior. For x > 0, y = 2 log x, which as x increases, y increases. For x < 0, y = 2 log(-x), which as x increases (from -∞ to 0), -x decreases from +∞ to 0, so log(-x) decreases, hence y decreases. So yes, decreasing on (-∞,0) and increasing on (0, +∞). Therefore, the monotonic decreasing interval is (-∞,0), which is option C. So the answer should be C.Final Answerboxed{C}</think>To determine the monotonic decreasing interval of the function ( y = lg x^2 ), we start by rewriting the function using logarithm properties. The function ( y = lg x^2 ) can be simplified to ( y = 2 log_{10} |x| ). This function is defined for all ( x neq 0 ).Next, we analyze the behavior of the function for ( x > 0 ) and ( x < 0 ) separately:1. For ( x > 0 ): - The function simplifies to ( y = 2 log_{10} x ). - The derivative is ( frac{dy}{dx} = frac{2}{x ln 10} ), which is positive because ( x > 0 ). Therefore, the function is increasing on ( (0, +infty) ).2. For ( x < 0 ): - The function simplifies to ( y = 2 log_{10} (-x) ). - The derivative is ( frac{dy}{dx} = frac{2}{x ln 10} ). Since ( x < 0 ), the derivative is negative. Therefore, the function is decreasing on ( (-infty, 0) ).Thus, the function ( y = lg x^2 ) is decreasing on the interval ( (-infty, 0) ) and increasing on ( (0, +infty) ). The monotonic decreasing interval is ( (-infty, 0) ).[boxed{C}],<think>Okay, so I need to figure out the monotonic decreasing interval for the function y = lg x². Let me start by recalling what a monotonic decreasing interval is. If a function is decreasing on an interval, then as x increases, the value of the function decreases. So, the slope or derivative should be negative in that interval. First, let me make sure I understand the function correctly. The function is y = lg x². Here, lg usually stands for logarithm base 10. So, y = log₁₀(x²). Hmm, right. Now, logarithm functions have certain domains. Since x² is inside the log, the argument x² must be positive. But x² is always positive except when x = 0, where it's zero. So, the domain of this function is all real numbers except x = 0. That's important because the function isn't defined at x = 0, so we have to consider intervals around that point.Now, the question is asking about where the function is decreasing. So, maybe I can find the derivative of y with respect to x and see where it's negative. Let's try that.First, rewrite the function to make differentiation easier. y = log₁₀(x²). Using logarithm properties, logₐ(b^c) = c logₐ b, so this becomes y = 2 log₁₀ |x|. Wait, actually, x² is the same as |x| squared, so log(x²) is 2 log|x|. Because x² = |x|², and log(a²) = 2 log|a|. That's correct. So, y = 2 log₁₀ |x|. That might make it easier to differentiate because the absolute value function has a known derivative.So, derivative of y with respect to x is dy/dx = 2 * derivative of log₁₀ |x|. Recall that the derivative of logₐ |x| is 1/(x ln a). So, derivative of log₁₀ |x| is 1/(x ln 10). Therefore, dy/dx = 2 * [1/(x ln 10)]. So, dy/dx = 2/(x ln 10). Now, to determine where the function is decreasing, we need dy/dx < 0. So, 2/(x ln 10) < 0. Let's solve this inequality. First, note that ln 10 is a positive number since 10 > 1. So, ln 10 ≈ 2.3026 > 0. The numerator here is 2, which is also positive. So, 2/(x ln 10) < 0. Since 2 and ln 10 are both positive, the sign of the entire expression depends on the sign of x. Therefore, 2/(x ln 10) < 0 implies that x must be negative. Because if x is positive, the entire expression is positive. If x is negative, then the denominator is negative, so the whole expression is negative.Therefore, the derivative is negative when x < 0. So, the function is decreasing on the interval where x < 0. But we need to remember that the function is defined for all real numbers except x = 0. So, the domain is (-∞, 0) ∪ (0, ∞). But the derivative is negative only when x is in (-∞, 0). Therefore, the function is decreasing on (-∞, 0). However, we need to check if it's decreasing on the entire interval or if there's any part where it might not be. Since the derivative is negative throughout (-∞, 0), the function is monotonically decreasing there. What about the interval (0, ∞)? On that interval, x is positive, so dy/dx = 2/(x ln 10) is positive. Therefore, the function is increasing on (0, ∞). So, the function is decreasing on (-∞, 0) and increasing on (0, ∞). But the question is asking for the monotonic decreasing interval. So, the correct interval is (-∞, 0). However, let me check the options given:A: R (all real numbers) – but the function isn't decreasing everywhere. It's decreasing only when x < 0 and increasing when x > 0, so this is incorrect.B: (-∞,0), (0,+∞) – this suggests that the function is decreasing on both intervals. But we just saw that on (0, ∞), the derivative is positive, so the function is increasing there. Therefore, B is incorrect.C: (-∞,0) – this is the interval where the function is decreasing. So, this is correct.D: (0,+∞) – this is where the function is increasing, so D is incorrect.Wait, but hold on. Let me just verify again. Maybe I made a mistake in the derivative?Original function: y = lg x². Let me compute dy/dx again. The derivative of log base 10 of x squared. Using chain rule: derivative of log(u) is 1/(u ln 10) * du/dx. So, u = x², du/dx = 2x. Therefore, dy/dx = (1/(x² ln 10)) * 2x = 2x/(x² ln 10) = 2/(x ln 10). Yes, that's the same as before. So, derivative is 2/(x ln 10). So, when x is positive, derivative is positive; when x is negative, derivative is negative. Therefore, function is decreasing for x < 0 and increasing for x > 0. Therefore, the answer is C: (-∞,0). But wait, the problem says "monotonic decreasing interval". Since the function is defined on two separate intervals, (-∞,0) and (0, ∞). On (-∞,0), the function is decreasing, so that's a monotonic decreasing interval. On (0, ∞), it's increasing. So, the correct answer should be C. However, let me check the options again. The options are:A: RB: (-∞,0), (0,+∞)C: (-∞,0)D: (0,+∞)The answer is C. So, the answer should be C.Wait, but let me think again. The problem says "monotonic decreasing interval". The term "interval" here might be a bit confusing because the function is defined on two separate intervals. But in calculus, when we talk about intervals of increase or decrease, we consider each interval where the function is defined. So, even though the domain is split into two intervals, we can still have each interval considered separately. But the options are given as choices. Option B is both intervals, but since the function is increasing on one and decreasing on the other, it's not decreasing on both. So, the only interval where it's decreasing is (-∞,0), which is option C. So, the answer is C.But wait, perhaps the problem is written in Chinese, and sometimes translations can be tricky. The original problem says "monotonic decreasing interval". In Chinese math terminology, when they ask for "monotonic decreasing interval", they might accept multiple intervals if applicable, but in this case, the function is decreasing only on (-∞,0). So, the answer is C.Alternatively, let's analyze the function's behavior without calculus. Let's consider y = log x². Since x² is symmetric about the y-axis, the function is even. So, it's symmetric. Let's take two points in (-∞,0), say x1 and x2 where x1 < x2 < 0. Then, x1² > x2² because as you go left on the negative side, squaring makes the number larger. Therefore, log x1² > log x2². So, as x increases from -∞ to 0, y decreases. Therefore, the function is decreasing on (-∞,0).Similarly, on the positive side, if we take x3 and x4 where 0 < x3 < x4, then x3² < x4², so log x3² < log x4². Therefore, as x increases from 0 to ∞, y increases. So, increasing there. Therefore, this confirms again that the decreasing interval is (-∞,0). Therefore, the answer is C.So, among the options given, C is correct.Final Answerboxed{C}</think>To determine the monotonic decreasing interval of the function ( y = lg x^2 ), we start by rewriting the function using logarithm properties: ( y = 2 lg |x| ). The domain of this function is all real numbers except ( x = 0 ).Next, we find the derivative of the function. Using the chain rule, the derivative of ( log_{10} |x| ) is ( frac{1}{x ln 10} ). Therefore, the derivative of ( y = 2 lg |x| ) is:[frac{dy}{dx} = 2 cdot frac{1}{x ln 10} = frac{2}{x ln 10}]To determine where the function is decreasing, we solve the inequality ( frac{2}{x ln 10} < 0 ). Since ( ln 10 ) is positive, the sign of the derivative depends on the sign of ( x ). The derivative is negative when ( x ) is negative. Thus, the function is decreasing on the interval ( (-infty, 0) ).For the interval ( (0, +infty) ), the derivative is positive, indicating the function is increasing there. Therefore, the monotonic decreasing interval of the function ( y = lg x^2 ) is ( (-infty, 0) ).[boxed{C}]]

🔑:AnalysisThis question examines the monotonicity of composite functions, paying attention to the domain of the function.To solve this, we should consider the monotonicity of the composite function in conjunction with the function's domain.SolutionThe domain of the function is clearly: (-infty,0)cup(0,+infty),Let u=x^{2}, then y=lg u is an increasing function on (0,+infty),And u=x^{2} is a decreasing function on (-infty,0) and an increasing function on (0,+infty).According to the monotonicity of the composite function, the decreasing interval of the original function should be (-infty,0).Therefore, the correct choice is boxed{C}.

🧠:[<think>Okay, let's try to tackle this problem step by step. So, the function given is f(x) = x³ + x, and we need to find the range of real number m such that the inequality f(m sinθ) + f(1 - m) > 0 holds for all θ in [0, π/2]. The answer choices are A to D. Hmm, let's start by understanding what's required here.First, since θ is between 0 and π/2, sinθ ranges from 0 to 1. Therefore, m sinθ will range from 0 to m, depending on the value of m. But m can be positive or negative, right? If m is positive, then m sinθ goes from 0 to m. If m is negative, then m sinθ goes from m to 0. Hmm, that's something to note.Now, the inequality is f(m sinθ) + f(1 - m) > 0 for all θ in [0, π/2]. So, we need to ensure that no matter what θ we pick in that interval, the sum of these two function evaluations is positive.Let's first recall what f(x) is. f(x) = x³ + x. Let's analyze this function. If we take its derivative, f’(x) = 3x² + 1, which is always positive since 3x² is non-negative and adding 1 makes it at least 1. Therefore, f(x) is strictly increasing on all real numbers. That's important because if f is strictly increasing, then for any a and b, if a < b then f(a) < f(b). So, the function is monotonic.Given that f is strictly increasing, maybe we can use that property to simplify the inequality. Let me see.So, the inequality is f(m sinθ) + f(1 - m) > 0. Let's consider breaking this down. Since sinθ can vary between 0 and 1, let's denote t = sinθ, where t ∈ [0,1]. Then the inequality becomes f(mt) + f(1 - m) > 0 for all t ∈ [0,1].So, now the problem is transformed into: Find all real numbers m such that f(mt) + f(1 - m) > 0 for all t ∈ [0,1]. That might be easier to handle.Since f is strictly increasing, if we can find the minimum value of f(mt) over t ∈ [0,1], then adding f(1 - m) to that minimum should still be greater than 0. Therefore, the minimal value of f(mt) would occur at the minimal or maximal t, depending on whether m is positive or negative.Let me think. Since m can be positive or negative, the function mt can be increasing or decreasing in t. Let's analyze both cases.Case 1: m > 0. Then, as t increases from 0 to 1, mt increases from 0 to m. Therefore, f(mt) is increasing in t because f is increasing. So, the minimum value of f(mt) occurs at t = 0, which is f(0) = 0³ + 0 = 0. Therefore, in this case, the minimal f(mt) is 0. Then, adding f(1 - m) to 0 should be greater than 0. Therefore, f(1 - m) > 0. Since f is strictly increasing, f(1 - m) > 0 implies 1 - m > f^{-1}(0). But f(0) = 0, so f^{-1}(0) = 0. Therefore, 1 - m > 0 ⇒ m < 1. So, if m is positive, we need m < 1. But m can be positive up to 1. But this is only for Case 1 where m > 0. So, combining these, m ∈ (0,1). But we have to check if this is sufficient.Wait, but hold on. If m is in (0,1), then 1 - m is positive, so f(1 - m) is positive because f(x) is positive when x is positive (since f(x) = x³ + x, which is x(x² + 1), and for x > 0, x³ + x > 0). Therefore, if m ∈ (0,1), then 1 - m ∈ (0,1), and f(1 - m) is positive, and the minimal f(m sinθ) is 0, so the sum is positive. So, that seems okay.But what if m is negative? Let's consider Case 2: m ≤ 0.If m is negative, then as t increases from 0 to 1, mt goes from m*0 = 0 to m*1 = m, but since m is negative, mt decreases from 0 to m. Therefore, mt ranges from m to 0. So, since f is increasing, the minimal value of f(mt) would be at the minimal mt, which is m*1 = m. Wait, but m is negative, so mt ∈ [m, 0], so the minimum of f(mt) would be at the left end, which is mt = m (when t = 1). Therefore, the minimal f(mt) is f(m). Therefore, in this case, f(m) + f(1 - m) > 0 must hold.So, for m ≤ 0, we require f(m) + f(1 - m) > 0. Let's compute that.f(m) + f(1 - m) = [m³ + m] + [(1 - m)³ + (1 - m)]Let's expand (1 - m)³: 1 - 3m + 3m² - m³.Therefore, f(m) + f(1 - m) = m³ + m + 1 - 3m + 3m² - m³ + 1 - mWait, wait. Let me compute step by step.First, f(m) = m³ + m.f(1 - m) = (1 - m)³ + (1 - m).Expand (1 - m)³: 1 - 3m + 3m² - m³.Therefore, f(1 - m) = 1 - 3m + 3m² - m³ + 1 - m.Wait, no. Wait, (1 - m)³ is 1 - 3m + 3m² - m³, then adding (1 - m) gives:1 - 3m + 3m² - m³ + 1 - m = 2 - 4m + 3m² - m³.Wait, let me compute f(1 - m):(1 - m)^3 + (1 - m) = (1 - 3m + 3m² - m³) + (1 - m) = 1 + 1 - 3m - m + 3m² - m³ = 2 - 4m + 3m² - m³.Therefore, f(m) + f(1 - m) = [m³ + m] + [2 - 4m + 3m² - m³] = m³ + m + 2 - 4m + 3m² - m³.Simplify terms:m³ - m³ cancels out.Then, m - 4m = -3m.So, remaining terms: 2 - 3m + 3m².Therefore, f(m) + f(1 - m) = 3m² - 3m + 2.So, the expression simplifies to 3m² - 3m + 2. Therefore, for m ≤ 0, we need 3m² - 3m + 2 > 0.But 3m² - 3m + 2 is a quadratic in m. Let's check if this quadratic is always positive.Compute discriminant D = (-3)^2 - 4*3*2 = 9 - 24 = -15. Since discriminant is negative, the quadratic has no real roots and opens upwards (coefficient of m² is 3 > 0). Therefore, 3m² - 3m + 2 > 0 for all real m. Therefore, for any real m, 3m² - 3m + 2 is positive. So, for m ≤ 0, the expression f(m) + f(1 - m) is always positive.Therefore, for m ≤ 0, the inequality holds for all θ ∈ [0, π/2].But in the case when m > 0, we saw earlier that we need m < 1. However, we need to check whether m in (0,1) actually satisfies the original inequality for all θ.Wait, when m ∈ (0,1), we said that the minimal value of f(m sinθ) is 0 (at θ = 0), so the sum f(0) + f(1 - m) = 0 + f(1 - m). Since 1 - m is in (0,1) when m ∈ (0,1), and f(1 - m) is positive because f(x) is positive for x > 0. Therefore, 0 + positive is positive, so the inequality holds.But wait, when θ = 0, sinθ = 0, so f(0) + f(1 - m) = 0 + (1 - m)^3 + (1 - m). Since 1 - m is between 0 and 1 when m ∈ (0,1), then (1 - m)^3 is positive, and (1 - m) is positive, so the sum is positive. Therefore, okay.But when θ approaches π/2, sinθ approaches 1, so m sinθ approaches m. Then, f(m) + f(1 - m). Wait, but since m ∈ (0,1), then 1 - m is still positive, so f(1 - m) is positive, and f(m) is positive as well because m is positive. So, the sum of two positive numbers is positive. Therefore, even at θ = π/2, the inequality holds.Wait, but hold on. For m ∈ (0,1), when θ = π/2, the term becomes f(m) + f(1 - m). But earlier, when we considered m ≤ 0, we saw that f(m) + f(1 - m) simplifies to 3m² - 3m + 2, which is always positive. But for m ∈ (0,1), let's compute f(m) + f(1 - m):f(m) + f(1 - m) = m³ + m + (1 - m)³ + (1 - m). Which is the same as 3m² - 3m + 2 as we saw earlier. Wait, but this is the same expression, regardless of m. Wait, that's interesting. So, regardless of m, f(m) + f(1 - m) = 3m² - 3m + 2. Which is always positive because discriminant is negative. So, even for m ∈ (0,1), f(m) + f(1 - m) is positive. Therefore, even at θ = π/2, where sinθ = 1, the expression f(m) + f(1 - m) is positive. So, in that case, the minimum over all θ is actually at θ = 0, where it's 0 + f(1 - m) = f(1 - m), which is positive, and at θ = π/2, it's f(m) + f(1 - m), which is also positive. So, in the interval (0,1), the inequality holds for all θ.But wait, but in our initial analysis for m > 0, we considered the minimal value of f(m sinθ) to be at θ = 0, which is 0, but actually, if m is positive, m sinθ ranges from 0 to m, so f(m sinθ) ranges from f(0) = 0 to f(m). Therefore, the minimal value is 0, but the expression f(m sinθ) + f(1 - m) ranges from 0 + f(1 - m) to f(m) + f(1 - m). But since f(1 - m) is positive and f(m) is positive, the entire expression is always positive. Therefore, for m ∈ (0,1), the inequality holds.But then, if m = 1, what happens? Let's check.If m = 1, then the expression becomes f(sinθ) + f(0). f(sinθ) is sin³θ + sinθ, and f(0) is 0. So, the expression is sin³θ + sinθ. Since sinθ ∈ [0,1], sin³θ + sinθ is always non-negative, but when θ = 0, it's 0 + 0 = 0. So, the inequality becomes 0 > 0, which is false. Therefore, m = 1 is not allowed.Similarly, if m > 1, then 1 - m is negative. Let's take m = 2. Then, for θ = π/2, sinθ = 1, so f(2*1) + f(1 - 2) = f(2) + f(-1). Compute f(2) = 8 + 2 = 10, f(-1) = (-1)^3 + (-1) = -1 -1 = -2. Therefore, 10 + (-2) = 8 > 0. But for θ = 0, f(0) + f(1 - 2) = 0 + f(-1) = -2 < 0. So, the inequality does not hold for m = 2. Therefore, m cannot be greater than or equal to 1.So, m must be less than 1. But in the case of m ∈ (0,1), we saw that the inequality holds. For m ≤ 0, the inequality also holds because even though f(m) is negative (since m ≤ 0, f(m) = m³ + m, which is m(m² + 1). Since m ≤ 0 and m² + 1 is always positive, f(m) is ≤ 0. However, f(1 - m) when m ≤ 0: since 1 - m ≥ 1, because m ≤ 0 ⇒ -m ≥ 0 ⇒ 1 - m ≥ 1. Then, f(1 - m) = (1 - m)^3 + (1 - m). Since 1 - m ≥ 1, this is a positive number. So, f(m) + f(1 - m) = [m³ + m] + [(1 - m)^3 + (1 - m)] = 3m² - 3m + 2, which we already determined is always positive. So, even though f(m) is negative, the sum is still positive.Therefore, combining both cases:- For m ≤ 0: The inequality holds.- For 0 < m < 1: The inequality holds.- For m ≥ 1: The inequality fails.Therefore, the range of m is (-∞, 1). But wait, the answer choices are A: (0,1), B: (-∞,0), C: (-∞,1/2), D: (-∞,1). So, D is (-∞,1). But according to our analysis, m can be any real number less than 1, so the correct answer should be D.Wait, but wait the answer choices include C: (-∞,1/2), but our analysis says up to 1. Maybe we need to check again. Let me verify.Wait, for m = 1/2, which is in (0,1), the inequality should hold. Let's test m = 1/2.So, f(m sinθ) + f(1 - m) = f((1/2) sinθ) + f(1 - 1/2) = f((1/2) sinθ) + f(1/2).Since f is increasing, f((1/2) sinθ) ≥ f(0) = 0, and f(1/2) = (1/2)^3 + 1/2 = 1/8 + 1/2 = 5/8. Therefore, the sum is ≥ 0 + 5/8 = 5/8 > 0. So, yes, it holds. Similarly, for m approaching 1 from below, say m = 0.9.Then, f(0.9 sinθ) + f(0.1). f(0.9 sinθ) ranges from f(0) = 0 to f(0.9) = 0.729 + 0.9 = 1.629. f(0.1) = 0.001 + 0.1 = 0.101. Therefore, the sum ranges from 0 + 0.101 = 0.101 > 0 to 1.629 + 0.101 = 1.73 > 0. So, it holds.But at m = 1, as previously checked, the sum is 0 when θ = 0, which is not greater than 0. So, m must be strictly less than 1.But the answer choice D is (-∞,1). But in the problem statement, it's stated that the inequality must hold for 0 ≤ θ ≤ π/2. If m is approaching 1 from below, then as m approaches 1, the minimal value of f(m sinθ) + f(1 - m) approaches f(0) + f(0) = 0 + 0 = 0. But since m is less than 1, 1 - m is approaching 0 from the positive side. Therefore, f(1 - m) is approaching f(0) = 0 from the positive side. So, even as m approaches 1, the minimal sum is approaching 0 but remains positive. Wait, no. Wait, for m approaching 1, 1 - m approaches 0. So, f(1 - m) = (1 - m)^3 + (1 - m). As 1 - m approaches 0, this is approximately (1 - m) + (1 - m) = 2(1 - m), which approaches 0. Then, f(m sinθ) + f(1 - m) when θ = 0 is 0 + f(1 - m) = f(1 - m), which approaches 0. So, as m approaches 1 from below, the expression approaches 0. But since m is strictly less than 1, 1 - m is strictly positive, so f(1 - m) is strictly positive. Therefore, even approaching 1, the inequality holds, but at m = 1, it fails. Therefore, the upper limit is m < 1, so the interval is (-∞,1). Hence, the correct answer is D.But wait, let me check m = 1/2, which is in option C. Our analysis shows that m can be up to 1, so why is option C present? Maybe the test makers thought something different. Wait, let's check another test case. Let m = 3/4.Then, f(m sinθ) + f(1 - m) = f((3/4) sinθ) + f(1/4). For θ = 0, this is 0 + f(1/4) = (1/4)^3 + 1/4 = 1/64 + 16/64 = 17/64 > 0. For θ = π/2, f(3/4) + f(1/4) = (27/64 + 3/4) + (1/64 + 1/4) = (27/64 + 48/64) + (1/64 + 16/64) = 75/64 + 17/64 = 92/64 = 23/16 > 0. So, even at m = 3/4, it's okay. So, why is option C (-∞,1/2) there? Maybe a miscalculation somewhere.Wait, let me check the expression again. When m > 0, we require that f(1 - m) > 0. But 1 - m must be greater than 0, which gives m < 1. So, as long as m <1, even m=0.9, f(1 - m)=f(0.1) >0. So, sum is positive. So, why would the answer be C? Hmm, maybe there's an error in my analysis.Wait, another approach. Since the function f is increasing and convex? Let's check the second derivative. f''(x) = 6x. For x ≥0, f is convex, and for x ≤0, concave. But I don't know if that helps here.Alternatively, maybe we can consider the inequality f(m sinθ) + f(1 - m) >0 for all θ ∈ [0, π/2], which can be rewritten as f(m sinθ) > -f(1 - m). Since f is strictly increasing, maybe we can invert this inequality.But f is invertible, so m sinθ > f^{-1}(-f(1 - m)). But since f is increasing, f^{-1}(y) is the value x such that f(x) = y. But I'm not sure if this helps.Alternatively, think of the minimum of f(m sinθ) over θ. Since sinθ ∈ [0,1], then m sinθ is between min(m*0, m*1) and max(m*0, m*1). Depending on the sign of m.If m >0, then m sinθ ∈ [0, m]. So, the minimal f(m sinθ) is f(0)=0.If m <0, then m sinθ ∈ [m,0]. So, the minimal f(m sinθ) is f(m).Thus, to ensure that f(m sinθ) + f(1 - m) >0 for all θ, we need:If m >0: 0 + f(1 - m) >0 ⇒ f(1 - m) >0 ⇒ 1 - m >0 ⇒ m <1.If m ≤0: f(m) + f(1 - m) >0.But as we calculated earlier, f(m) + f(1 - m) =3m² -3m +2, which is always positive for all real m because discriminant is negative. So, for m ≤0, the inequality holds.Therefore, combining both cases, m <1. So, the range is (-∞,1), which is option D.But the answer options don't include option D as (−∞,1), but looking back, the options are:A: (0,1)B: (−∞,0)C: (−∞,1/2)D: (−∞,1)So, D is indeed (−∞,1). Therefore, the correct answer is D. So, why did I think maybe C? Because perhaps miscalculating something. But according to the analysis, the answer is D.Therefore, the answer should be D.Final Answerboxed{D}</think>Given the function ( f(x) = x^3 + x ), we need to find the range of real number ( m ) such that the inequality ( f(m sin theta) + f(1 - m) > 0 ) holds for all ( theta in [0, frac{pi}{2}] ).1. Analyzing the function ( f(x) ): - ( f(x) ) is strictly increasing since its derivative ( f'(x) = 3x^2 + 1 ) is always positive. - ( f(x) ) is positive for ( x > 0 ) and negative for ( x < 0 ).2. Case analysis for ( m ): - Case 1: ( m > 0 ): - For ( theta in [0, frac{pi}{2}] ), ( m sin theta ) ranges from 0 to ( m ). - The minimum value of ( f(m sin theta) ) occurs at ( theta = 0 ), which is ( f(0) = 0 ). - The inequality becomes ( 0 + f(1 - m) > 0 ). For this to hold, ( 1 - m > 0 Rightarrow m < 1 ). - Therefore, ( m in (0, 1) ). - Case 2: ( m leq 0 ): - For ( theta in [0, frac{pi}{2}] ), ( m sin theta ) ranges from ( m ) to 0. - The minimum value of ( f(m sin theta) ) occurs at ( theta = frac{pi}{2} ), which is ( f(m) ). - The inequality becomes ( f(m) + f(1 - m) > 0 ). - Simplifying ( f(m) + f(1 - m) ): [ f(m) + f(1 - m) = m^3 + m + (1 - m)^3 + (1 - m) ] [ = m^3 + m + 1 - 3m + 3m^2 - m^3 + 1 - m ] [ = 3m^2 - 3m + 2 ] - The quadratic ( 3m^2 - 3m + 2 ) has a negative discriminant (( -15 )), ensuring it is always positive for all real ( m ).3. Conclusion: - For ( m > 0 ), the valid range is ( (0, 1) ). - For ( m leq 0 ), the inequality holds for all ( m ). - Combining both cases, the range of ( m ) is ( (-infty, 1) ).Thus, the correct answer is (boxed{D}).,<think>Okay, let me try to figure this problem out. So, we have the function f(x) = x³ + x, and we need to find the range of real numbers m such that the inequality f(m sinθ) + f(1 - m) > 0 holds for all θ in [0, π/2]. The answer choices are given as A, B, C, D. Alright, let's break this down step by step.First, let me recall that f(x) = x³ + x. Let me note that this function is a cubic function. Since the coefficient of x³ is positive, the function tends to positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. But more importantly, since the derivative f’(x) = 3x² + 1, which is always positive (because 3x² is non-negative and adding 1 makes it at least 1), this means f(x) is strictly increasing over all real numbers. That's a crucial point because if the function is strictly increasing, then inequalities can be directly translated to their arguments.So, if f(a) + f(b) > 0, then since f is strictly increasing, but adding two functions complicates things a bit. However, maybe we can manipulate the inequality f(m sinθ) + f(1 - m) > 0 to find a condition on m.Given that θ is between 0 and π/2, sinθ ranges from 0 to 1. Therefore, m sinθ will range from 0 to m if m is positive, or from m to 0 if m is negative. So, depending on the sign of m, the argument of f in the first term will vary.But maybe instead of dealing with θ directly, let's consider the range of m sinθ. Let's denote t = sinθ. Since θ ∈ [0, π/2], t ∈ [0, 1]. So, the problem reduces to: For all t ∈ [0,1], f(mt) + f(1 - m) > 0. We need to find m such that this inequality holds for all t in [0,1].So, restated: For all t ∈ [0,1], ( (mt)^3 + mt ) + ( (1 - m)^3 + (1 - m) ) > 0.Let's write that out:(m³ t³ + mt) + ( (1 - m)^3 + (1 - m) ) > 0.Combine the terms:m³ t³ + mt + (1 - m)^3 + (1 - m) > 0.Hmm. That's the inequality we need to satisfy for all t in [0,1]. Let's see if we can analyze this expression.Let me consider grouping terms differently. Let's separate the terms involving m and constants.First, let's compute (1 - m)^3 + (1 - m):(1 - m)^3 + (1 - m) = (1 - 3m + 3m² - m³) + (1 - m) = 1 - 3m + 3m² - m³ + 1 - m = 2 - 4m + 3m² - m³.Similarly, the terms involving t are m³ t³ + mt. Let me factor out m t:m t (m² t² + 1). Hmm, not sure if that's helpful. Alternatively, note that m³ t³ + mt = mt(m² t² + 1). Since t ∈ [0,1], and m is a real number, perhaps we can find the minimum value of this expression over t ∈ [0,1], and then set the entire expression to be greater than 0.So, the total expression is:mt(m² t² + 1) + (2 - 4m + 3m² - m³) > 0.But maybe this approach is getting too complicated. Let's think differently.Since f is strictly increasing, we might be able to find some inequalities based on the range of m sinθ. Let's recall that f is increasing, so if we can bound m sinθ in some interval, then we can bound f(m sinθ). But since we need the sum f(m sinθ) + f(1 - m) to be positive for all θ, maybe we can consider the minimum value of f(m sinθ) over θ ∈ [0, π/2], add f(1 - m), and require that the result is positive.Given that f is increasing, the minimum value of f(m sinθ) occurs at the minimum value of m sinθ. But the minimum of m sinθ depends on the sign of m.If m is positive, then sinθ ranges from 0 to 1, so m sinθ ranges from 0 to m. Since f is increasing, the minimum value of f(m sinθ) is f(0) = 0³ + 0 = 0.If m is negative, then m sinθ ranges from m*1 to m*0, i.e., from m to 0. Again, since f is increasing, the minimum value of f(m sinθ) is f(m).Therefore, depending on the sign of m, the minimum of f(m sinθ) is different.So, let's consider two cases: m ≥ 0 and m < 0.Case 1: m ≥ 0.In this case, m sinθ ∈ [0, m], so f(m sinθ) ∈ [f(0), f(m)]. The minimum value of f(m sinθ) is f(0) = 0. Therefore, the inequality becomes 0 + f(1 - m) > 0 ⇒ f(1 - m) > 0. Since f is increasing, f(1 - m) > 0 ⇨ 1 - m > f^{-1}(0). Let's find f^{-1}(0). f(x) = 0 ⇒ x³ + x = 0 ⇒ x(x² + 1) = 0 ⇒ x = 0. Therefore, f(x) > 0 ⇨ x > 0. So, f(1 - m) > 0 ⇨ 1 - m > 0 ⇨ m < 1.But in this case, m ≥ 0, so combining m ≥ 0 and m < 1 gives 0 ≤ m < 1.However, the problem states "the inequality always holds", so for m in [0,1), the inequality f(m sinθ) + f(1 - m) > 0 would hold, because the minimum f(m sinθ) is 0, and f(1 - m) > 0.But wait, in the case m = 0, let's check. If m = 0, then f(m sinθ) = f(0) = 0, and f(1 - 0) = f(1) = 1 + 1 = 2. So 0 + 2 = 2 > 0, which is true. So m = 0 is allowed? But the answer choices don't include 0 in all options. The options are A: (0,1), B: (-∞,0), C: (-∞,1/2), D: (-∞,1). Wait, but if m can be in [0,1), then the range would include [0,1), but none of the options include 0 except possibly D which is (-∞,1). But D is up to 1, but not including 1. Wait, the answer choices are A: (0,1); B: (-∞,0); C: (-∞,1/2); D: (-∞,1). So, the case m ≥ 0 gives us m ∈ [0,1). But if m is allowed to be negative, we need to check that case too.Case 2: m < 0.In this case, m sinθ ranges from m to 0 (since sinθ ∈ [0,1], multiplying by negative m gives [m, 0]). Since f is increasing, the minimum value of f(m sinθ) is f(m). Therefore, the inequality becomes f(m) + f(1 - m) > 0.So, in this case, we need f(m) + f(1 - m) > 0. Let's compute this:f(m) + f(1 - m) = [m³ + m] + [(1 - m)^3 + (1 - m)].Let's expand (1 - m)^3:(1 - m)^3 = 1 - 3m + 3m² - m³.Therefore, f(m) + f(1 - m) = m³ + m + 1 - 3m + 3m² - m³ + 1 - m.Simplify term by term:m³ - m³ = 0.m - 3m - m = -3m.1 + 1 = 2.3m².So, total expression is 3m² - 3m + 2.So, f(m) + f(1 - m) = 3m² - 3m + 2.We need this to be greater than 0 for m < 0.So, the inequality is 3m² - 3m + 2 > 0.Let's analyze this quadratic in m.First, compute the discriminant: D = (-3)^2 - 4*3*2 = 9 - 24 = -15.Since the discriminant is negative, the quadratic 3m² - 3m + 2 is always positive for all real m. Therefore, for m < 0, f(m) + f(1 - m) = 3m² - 3m + 2 > 0 always holds. Therefore, for m < 0, the inequality is always satisfied.So, combining both cases:Case 1: m ≥ 0: m < 1.Case 2: m < 0: always true.Therefore, the total range of m is (-∞,1). But wait, in Case 1, we have m ∈ [0,1). But combining with Case 2, m < 1. So overall, m ∈ (-∞,1). Let me check m =1. If m =1, then f(m sinθ) + f(1 - m) = f(sinθ) + f(0). Since f(sinθ) = sin³θ + sinθ ≥ 0 (since sinθ ∈ [0,1]), and f(0) =0. Therefore, the sum is ≥ 0. But the problem says the inequality must be >0 always. So when m =1, the sum is ≥0, but not strictly greater than 0. For example, when θ =0, sinθ =0, so f(0) + f(0) =0 +0=0, which is not greater than 0. Therefore, m=1 is excluded. Therefore, m must be less than 1.Similarly, check m=1/2. If m=1/2, then in case 1, m is between 0 and1. Then f(1 - m)=f(1 -1/2)=f(1/2)= (1/2)^3 +1/2=1/8 +1/2=5/8>0. Therefore, for m=1/2, the minimum of f(m sinθ) is 0, so the sum is 0 +5/8>0. So that's okay.But the answer choices include C: (-∞,1/2). But according to our analysis, m can be up to 1. But the answer D is (-∞,1). Wait, but according to the case analysis, m can be less than 1. However, let's check if there's a mistake here.Wait, perhaps my reasoning for m ≥0 is incomplete. Let me double-check.For m ≥0, the minimum of f(m sinθ) is f(0)=0. Therefore, the sum f(m sinθ) + f(1 - m) ≥0 + f(1 -m). Therefore, to have the sum >0 for all θ, we must have f(1 -m) >0. Since f is strictly increasing, f(1 -m) >0 ⇨1 -m >0 ⇨m <1. So, for m ≥0, m must be in [0,1). For m <0, we saw that regardless of m, the sum f(m sinθ) + f(1 -m) is equal to 3m² -3m +2, which is always positive. Therefore, the total range is m <1. So, the answer is D: (-∞,1).But let's check the answer options. The answer D is (-∞,1). The answer C is (-∞,1/2). Let me check whether m=3/4, which is in (-∞,1), satisfies the condition. So take m=3/4. Then for m ≥0, check if f(1 -m)=f(1 -3/4)=f(1/4)= (1/4)^3 +1/4=1/64 +16/64=17/64>0. So then f(m sinθ)+f(1 -m) ≥0 +17/64>0. So m=3/4 is okay.But wait, if m=1/2, which is in C, but m=3/4 is not in C but in D. So why is the answer not D? Wait, according to the answer choices, D is (-∞,1). So according to our analysis, that's correct.But let's check with m approaching 1 from below. Let m=1 - ε where ε is a small positive number. Then f(1 -m)=f(ε)=ε³ +ε>0. So even as ε approaches 0, f(ε) approaches 0, but is still positive. Therefore, the sum f(m sinθ)+f(1 -m) will approach 0 +0=0 as ε approaches 0. Wait, but m is approaching 1 from below. So m=1 - ε, so 1 -m=ε. Then, f(1 -m)=f(ε)=ε³ +ε. As ε approaches 0, this approaches 0. Therefore, even though f(1 -m) >0, the minimum of f(m sinθ) is 0 (since m is positive), so the sum approaches 0. Therefore, for m approaching 1 from below, the sum can be made arbitrarily small positive. But as long as m <1, the sum is still positive. So even though the sum approaches 0, it's still greater than 0. Therefore, m can be up to but not including 1. Therefore, the range is all m <1, which is option D.But why is there an option C? Maybe the problem is more complex, and my initial analysis is missing something.Wait, let's check for m=1/2. Suppose m=1/2. Then, according to the expression in case 2, when m is positive, the key is that f(1 -m) must be positive. For m=1/2, 1 -m=1/2>0, so f(1/2)= (1/2)^3 +1/2=1/8 +4/8=5/8>0. Therefore, the sum is always at least 5/8>0. Therefore, m=1/2 is acceptable.Wait, but maybe there is a mistake in case 2. Wait, when m <0, we said that f(m sinθ) has a minimum at f(m), since m sinθ ∈ [m,0], so the minimum is f(m). Then, adding f(1 -m), which for m <0, 1 -m >1, since m is negative. Then, f(1 -m) = (1 -m)^3 + (1 -m). Since 1 -m >1, and f is increasing, f(1 -m) > f(1) =1 +1=2. But also, f(m) is m³ +m. For m negative, m³ is negative, and m is negative, so f(m) is negative. However, when we computed f(m) +f(1 -m), we found that it equals 3m² -3m +2, which is a quadratic in m with a positive leading coefficient and discriminant negative, so always positive. Therefore, even though f(m) is negative and f(1 -m) is positive, their sum is always positive for m <0.Therefore, for m <0, the inequality holds. For m ∈ [0,1), the inequality holds because f(1 -m) >0. At m=1, the inequality fails because f(1 -1)=f(0)=0 and f(m sinθ)=f(sinθ) ≥0, but when θ=0, sinθ=0, so f(0)+f(0)=0. Therefore, m=1 is invalid. For m>1, 1 -m is negative, so f(1 -m) = (1 -m)^3 + (1 -m). Since 1 -m is negative, and f is an odd function plus something? Wait, f(x)=x³ +x. If x is negative, f(x) is x³ +x = -( |x|³ + |x| ). So f(x) is negative when x is negative. Therefore, if m>1, then 1 -m is negative, so f(1 -m) is negative. Then, f(m sinθ) is also positive because m sinθ is positive (since m>1 and sinθ ≥0). Therefore, the sum f(m sinθ) + f(1 -m) is a positive plus a negative. Depending on the magnitude, it might not be positive for all θ.For example, take m=2. Then, f(2 sinθ) + f(1 -2)=f(2 sinθ) +f(-1). Compute f(2 sinθ)=8 sin³θ + 2 sinθ. Since sinθ ∈ [0,1], this ranges from 0 to 8 + 2=10. f(-1)=(-1)^3 + (-1)= -1 -1= -2. Therefore, the sum ranges from 0 + (-2)= -2 to 10 + (-2)=8. So for m=2, the sum can be negative (e.g., when θ=0, sum is -2 <0). Therefore, m=2 is invalid. Therefore, m>1 are invalid.Similarly, take m=1.5. Then, f(1.5 sinθ) + f(1 -1.5)=f(1.5 sinθ) +f(-0.5). Compute f(1.5 sinθ)= (1.5)^3 sin³θ +1.5 sinθ ≈3.375 sin³θ +1.5 sinθ. The minimum value when sinθ=0 is 0. f(-0.5)= (-0.5)^3 + (-0.5)= -0.125 -0.5= -0.625. Therefore, the sum when θ=0 is 0 + (-0.625)= -0.625 <0. Hence, m=1.5 is invalid.Therefore, m must be less than 1. So overall, combining both cases, m ∈ (-∞,1). So the answer should be D.But the answer choices include C: (-∞,1/2). Maybe there is a mistake in my analysis.Wait, let's check m=0.5, which is in option C. If m=0.5, then for θ ∈ [0,π/2], sinθ ∈ [0,1], so m sinθ ∈ [0,0.5]. Then, f(m sinθ)= (0.5 sinθ)^3 +0.5 sinθ. The minimum is 0, when sinθ=0. Then f(1 -0.5)=f(0.5)=0.125 +0.5=0.625>0. So the sum is 0 +0.625=0.625>0. So okay. For m=0.75, which is not in C but in D. Let's check. f(m sinθ)= (0.75 sinθ)^3 +0.75 sinθ. The minimum is still 0. Then, f(1 -0.75)=f(0.25)=0.015625 +0.25≈0.2656>0. Therefore, sum is ≈0.2656>0. So m=0.75 is okay.But what about m approaching 1 from below. Take m=0.99. Then f(1 -0.99)=f(0.01)=0.000001 +0.01≈0.010001>0. So the sum is ≈0.010001>0. Hence, even as m approaches 1, the sum is positive. Therefore, m can be up to 1, but not including 1. So the correct range is (-∞,1), which is option D.But why is there option C? Maybe there is a miscalculation here.Alternatively, maybe I made a mistake in the case when m is negative. Let's check m negative. Suppose m=-1. Then f(m sinθ) + f(1 -m) = f(-sinθ) + f(2). Compute f(-sinθ)= (-sinθ)^3 + (-sinθ)= -sin³θ - sinθ. f(2)=8 +2=10. Therefore, the sum is (-sin³θ - sinθ) +10=10 - sinθ - sin³θ. Since sinθ ∈ [0,1], sinθ + sin³θ ∈ [0,1 +1]=2. Therefore, the sum is ≥10 -2=8>0. So m=-1 is okay.Take m=-0.5. Then f(-0.5 sinθ) +f(1 -(-0.5))=f(-0.5 sinθ) +f(1.5). Compute f(-0.5 sinθ)= (-0.5 sinθ)^3 + (-0.5 sinθ)= -0.125 sin³θ -0.5 sinθ. f(1.5)=3.375 +1.5=4.875. So sum is -0.125 sin³θ -0.5 sinθ +4.875. Since sinθ ∈ [0,1], the negative terms are at most -0.125 -0.5= -0.625, so sum ≥4.875 -0.625=4.25>0. Therefore, m=-0.5 is okay.Therefore, even for m negative, the sum is positive. Therefore, combining all m <1, the answer is D.Hence, the correct answer should be D.Final Answerboxed{D}</think>Given the function ( f(x) = x^3 + x ), we need to determine the range of the real number ( m ) such that the inequality ( f(m sin theta) + f(1 - m) > 0 ) holds for all ( 0 leqslant theta leqslant frac{pi}{2} ).1. Function Analysis: - The function ( f(x) = x^3 + x ) is strictly increasing because its derivative ( f'(x) = 3x^2 + 1 ) is always positive. - For ( 0 leqslant theta leqslant frac{pi}{2} ), ( sin theta ) ranges from 0 to 1. Therefore, ( m sin theta ) ranges from 0 to ( m ) if ( m geq 0 ), and from ( m ) to 0 if ( m < 0 ).2. Case Analysis: - Case 1: ( m geq 0 ): - The minimum value of ( f(m sin theta) ) occurs at ( sin theta = 0 ), which is ( f(0) = 0 ). - The inequality simplifies to ( f(1 - m) > 0 ). Since ( f ) is strictly increasing, ( 1 - m > 0 implies m < 1 ). - Therefore, ( 0 leq m < 1 ). - Case 2: ( m < 0 ): - The minimum value of ( f(m sin theta) ) occurs at ( sin theta = 1 ), which is ( f(m) ). - The inequality becomes ( f(m) + f(1 - m) > 0 ). Expanding this, we get: [ f(m) + f(1 - m) = (m^3 + m) + ((1 - m)^3 + (1 - m)) ] Simplifying, we find: [ 3m^2 - 3m + 2 ] - The quadratic ( 3m^2 - 3m + 2 ) has a negative discriminant, ensuring it is always positive for all ( m ).3. Combining Cases: - For ( m geq 0 ), the valid range is ( 0 leq m < 1 ). - For ( m < 0 ), the inequality always holds. - Combining both cases, the range of ( m ) is ( (-infty, 1) ).Thus, the range of the real number ( m ) is (boxed{D}).]

🔑:AnalysisThis problem examines the odd and even properties and monotonicity of functions, the transformation of ideas, and the issue of constant validity. By utilizing the property that the odd function f(x)=x^{3}+x is monotonically increasing, the inequality f(msin theta) + f(1-m) > 0 can be transformed into m(1-sin theta) 0, indicating that the function f(x) is monotonically increasing over mathbb{R}, hence, f(msin theta) + f(1-m) > 0 always holds,which can be transformed into: f(msin theta) > -f(1-m) = f(m-1) always holds, thus, msin theta > m-1 always holds for 0 leqslant theta leqslant frac{pi}{2},leading to m(1-sin theta) < 1 always holding,i.e., m < frac{1}{1-sin theta} always holds, given 0 leqslant sin theta leqslant 1,we have 0 leqslant 1-sin theta leqslant 1,when 1-sin theta=0, the inequality always holds,when 1-sin theta neq 0, frac{1}{1-sin theta} in [1,+infty),since m < frac{1}{1-sin theta} always holds,thus, m < 1, therefore, the range of the real number m is (-infty,1)．Hence, the correct choice is boxed{text{D}}.

🧠:[<think>Okay, so I need to solve this inequality system and figure out which option is correct. The problem gives me two inequalities:1. x - 3 < 22. 3x + 1 ≥ 2xAnd the options are A through D. Let me take this step by step.First, I'll handle the first inequality: x - 3 < 2. To solve for x, I can add 3 to both sides. That should isolate the x term. Let me do that:x - 3 + 3 < 2 + 3 x < 5So, the first inequality simplifies to x is less than 5. Alright, that's straightforward. Now, moving on to the second inequality: 3x + 1 ≥ 2x. Hmm, let's see. I need to get all the x terms on one side. Maybe subtract 2x from both sides?3x + 1 - 2x ≥ 2x - 2x x + 1 ≥ 0Wait, that simplifies to x + 1 ≥ 0. Then, subtracting 1 from both sides gives:x ≥ -1So, the second inequality simplifies to x is greater than or equal to -1. Now, combining both inequalities, the solution set should satisfy both conditions. So, x has to be greater than or equal to -1 and less than 5. Let me write that down:-1 ≤ x < 5Looking at the options, option C is "-1 ≤ x < 5", which matches what I found. Let me check the options again to make sure.A: x < 5 – That only considers the first inequality but ignores the second one. So, if x is, say, -2, which is less than 5, it would satisfy A but not the second inequality. So A is too broad.B: 1 ≤ x < 5 – Wait, the second inequality was x ≥ -1, not 1. So this lower bound is incorrect. Maybe a mistake in solving the second inequality? Let me check again.Original second inequality: 3x + 1 ≥ 2x. Subtract 2x from both sides: x + 1 ≥ 0. Then subtract 1: x ≥ -1. Yeah, so B is wrong because it starts at 1 instead of -1.C: -1 ≤ x < 5 – That's exactly what I got. So that's the combined solution set.D: x ≤ -1 – That contradicts the first inequality, because if x is less than or equal to -1, then x is certainly less than 5, but the second inequality requires x ≥ -1. So x can't be less than -1. So D is incorrect.So, the correct answer should be option C. Let me double-check to make sure I didn't make any mistakes in my algebra.First inequality: x - 3 < 2 Add 3: x < 5. Correct.Second inequality: 3x + 1 ≥ 2x Subtract 2x: x + 1 ≥ 0 Subtract 1: x ≥ -1. Correct.Combined: x has to be ≥ -1 and < 5. So, interval notation: [-1, 5). Which is exactly option C.Therefore, the answer is C.Final Answerboxed{C}</think>To solve the given system of inequalities, we start by solving each inequality separately.1. For the first inequality ( x - 3 < 2 ): [ x - 3 < 2 implies x < 2 + 3 implies x < 5 ]2. For the second inequality ( 3x + 1 geq 2x ): [ 3x + 1 geq 2x implies 3x - 2x + 1 geq 0 implies x + 1 geq 0 implies x geq -1 ]Combining the solutions of both inequalities, we need ( x ) to satisfy both ( x < 5 ) and ( x geq -1 ). Therefore, the solution set is:[-1 leq x < 5]The correct answer is (boxed{C}).,<think>Okay, so I need to solve this inequality system and figure out which option is correct. Let me start by looking at each inequality one by one. The system is:[begin{cases}x - 3 < 2 3x + 1 geq 2xend{cases}]Alright, first inequality: x - 3 < 2. Hmm, to solve for x, I can add 3 to both sides. Let me do that.x - 3 + 3 < 2 + 3 x < 5Okay, so the first inequality simplifies to x is less than 5. Got that. So that's one part of the solution set.Now the second inequality: 3x + 1 ≥ 2x. Let me solve this one. I should get all the x terms on one side. Subtract 2x from both sides:3x - 2x + 1 ≥ 0 x + 1 ≥ 0Hmm, wait, is that right? Let me check again. 3x +1 - 2x ≥ 0? So combining like terms: (3x - 2x) +1 ≥ 0, which is x +1 ≥ 0. Yeah, that's correct. Then subtract 1 from both sides:x ≥ -1So the second inequality simplifies to x is greater than or equal to -1. So putting both inequalities together, the solution set is all x such that x ≥ -1 and x < 5. So in interval notation, that's [-1, 5). Let me check the answer options:A: x < 5 (this is only the first inequality, not considering the second one) B: 1 ≤ x < 5 (Hmm, lower bound is 1? But the second inequality was x ≥ -1. Did I do something wrong here?) C: -1 ≤ x <5 (Yes, that's what I found) D: x ≤ -1 (That's not right, it's the opposite)Wait, the options are A, B, C, D. So based on my solution, it should be C. But wait, why is option B there with 1? Let me check my steps again.First inequality: x -3 <2: x <5. Correct. Second inequality: 3x +1 ≥2x. Subtract 2x: x +1 ≥0. Subtract 1: x ≥-1. So yes, that's correct. So the solution is x between -1 (inclusive) and 5 (exclusive). Therefore, option C. But let me see why B is 1. Maybe I made a mistake? Let me double-check the second inequality.3x +1 ≥2x. If I subtract 3x from both sides: 1 ≥ -x. Then multiply both sides by -1, which reverses the inequality: x ≥ -1. Yeah, same result. So that's correct. So the lower bound is -1, not 1. So option B is wrong. Option C is correct.Therefore, the answer is C.Final Answerboxed{C}</think>To solve the inequality system (left{begin{array}{l}x - 3 < 2 3x + 1 geq 2xend{array}right.):1. Solve the first inequality (x - 3 < 2): [ x - 3 < 2 implies x < 5 ]2. Solve the second inequality (3x + 1 geq 2x): [ 3x + 1 geq 2x implies 3x - 2x + 1 geq 0 implies x + 1 geq 0 implies x geq -1 ]Combining the solutions of both inequalities, we get:[-1 leq x < 5]Thus, the solution set is (boxed{C}).]

🔑:To solve the given system of inequalities, we'll tackle each inequality step by step.For the first inequality:[begin{align*}x - 3 &< 2 x &< 2 + 3 x &< 5 quad text{(①)}end{align*}]For the second inequality:[begin{align*}3x + 1 &geq 2x 3x - 2x &geq -1 x &geq -1 quad text{(②)}end{align*}]Combining the solutions from ① and ②, we find that the solution set for the system of inequalities is the intersection of the two solutions, which is -1 leqslant x < 5.Therefore, the correct answer is boxed{C}.

🧠:[<think>Okay, so I have this problem here: If non-zero vectors α and β satisfy |α + β| = |α − β|, then what is the angle between α and β? Hmm, let's see. I need to figure out the angle between two vectors given that the magnitude of their sum is equal to the magnitude of their difference. Alright, I remember from my vector classes that when dealing with magnitudes of vectors, the dot product comes into play. Maybe I can use that here.First, let me recall the formula for the magnitude of a vector. The magnitude of a vector, say |γ|, is given by the square root of the dot product of γ with itself. So, |γ|² = γ · γ. That might be helpful here. Let me try squaring both sides of the equation given in the problem to eliminate the square roots. That should simplify things a bit.The problem states that |α + β| = |α − β|. If I square both sides, I get |α + β|² = |α − β|². Now, let's compute each side using the dot product. Starting with the left side: |α + β|². Expanding this, it should be (α + β) · (α + β). Similarly, the right side is |α − β|² = (α − β) · (α − β). Let me compute both expansions.Left side: (α + β) · (α + β) = α · α + α · β + β · α + β · β. Since the dot product is commutative (α · β = β · α), this simplifies to |α|² + 2(α · β) + |β|².Right side: (α − β) · (α − β) = α · α − α · β − β · α + β · β. Again, using commutativity, this becomes |α|² − 2(α · β) + |β|².So now we have |α|² + 2(α · β) + |β|² = |α|² − 2(α · β) + |β|². Let me subtract |α|² and |β|² from both sides to simplify further. That leaves us with 2(α · β) = -2(α · β). Hmm, if I bring all terms to one side, I get 2(α · β) + 2(α · β) = 0. That simplifies to 4(α · β) = 0. Dividing both sides by 4, we find that α · β = 0. Wait, so the dot product of α and β is zero? That means the vectors are orthogonal! So, the angle between them must be 90 degrees, or π/2 radians. But let me confirm this step by step to make sure I didn't make any mistakes.Starting again: |α + β| = |α − β|. Squaring both sides gives the squared magnitudes equal. Expanding both sides, we get the sum of squares plus twice the dot product on the left, and sum of squares minus twice the dot product on the right. Setting them equal cancels out the sum of squares, leading to 2(α · β) = -2(α · β), which gives 4(α · β) = 0. Therefore, α · β = 0.Yes, that seems correct. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them: α · β = |α||β|cosθ. If this is zero, then either |α| or |β| is zero, but the problem states they are non-zero vectors. Therefore, cosθ must be zero. So, cosθ = 0. The angle θ between 0 and π radians (or 0 to 180 degrees) where cosine is zero is θ = π/2 radians, which is 90 degrees. Therefore, the angle between α and β is 90 degrees. Let me see if there's another way to approach this, maybe geometrically. If the magnitude of the sum and the difference of two vectors are equal, then the vectors form a rhombus when added and subtracted, since both the sum and difference vectors have the same length. In a rhombus, the diagonals are perpendicular. Wait, no, actually, in a rhombus, the diagonals are not necessarily perpendicular unless it's a square. Hmm, maybe that's not the right analogy.Alternatively, when vectors are such that |α + β| = |α − β|, they form a rectangle? Let me think. If you imagine vectors α and β as adjacent sides of a parallelogram, then α + β and α − β are the diagonals. If the diagonals are equal in length, then the parallelogram is a rectangle. Because in a rectangle, the diagonals are equal. Therefore, if the diagonals of a parallelogram are equal, the parallelogram is a rectangle, which implies that the original vectors are perpendicular. So that's another way to see it. Therefore, the angle between α and β is 90 degrees.So both algebraically and geometrically, the conclusion is the same. The angle between the vectors is π/2. Therefore, the answer should be 90 degrees or π/2 radians. The problem asks for the angle, so depending on the required format, but since they didn't specify, either should be acceptable, but in math problems like this, radians are often preferred, especially if no degree symbol is used. However, since the problem doesn't specify, maybe we should check standard conventions. In vector problems, angles are often expressed in degrees unless stated otherwise, but in calculus and higher mathematics, radians are standard. Hmm. The problem is in Chinese, but the instruction here is in English, so maybe safest to use both? Wait, no, the answer should be boxed. The original problem probably expects degrees or radians. Let me check.Wait, looking at the problem again: "the angle between α and β is". It doesn't specify, but in such cases, sometimes it's in radians, sometimes in degrees. However, since in mathematical analysis, angles are typically measured in radians unless specified. But in problems involving basic vectors, degrees are common. However, since π/2 is a common angle in radians, it's possible they expect that. Alternatively, maybe the answer is 90°, but to check, let's see the initial equation.Alternatively, perhaps the problem is from a linear algebra context, where angles are often in radians. But the problem itself is pretty fundamental. Let me see. If the answer is π/2, then boxed as boxed{dfrac{pi}{2}}. If degrees, then boxed{90^circ}. The question is in Chinese, but the instructions here are in English. In Chinese textbooks, degrees are commonly used unless specified otherwise. Hmm. But maybe to check, in the calculation, we showed that the cosine of the angle is zero, leading to θ = arccos(0) = π/2 radians or 90 degrees.But since the problem is about vectors, which can be in any dimension, and the angle is a geometric concept, perhaps both are acceptable. However, in mathematical terms, unless specified, angles are in radians. But sometimes in physics problems, they are in degrees. Since the problem is purely mathematical, likely radians. But to be safe, perhaps check similar problems. For example, in the classic problem where if vectors are perpendicular, their dot product is zero, and the angle is π/2. So, in that case, the answer is π/2. Therefore, I think the answer is π/2 radians.Therefore, the angle between α and β is π/2, which is 90 degrees. So, depending on the required answer, but since the problem didn't specify, but in mathematical contexts, radians are standard. So, boxed as boxed{dfrac{pi}{2}}.Final AnswerThe angle between α and β is boxed{dfrac{pi}{2}}.</think>Given non-zero vectors (boldsymbol{alpha}) and (boldsymbol{beta}) such that (|boldsymbol{alpha} + boldsymbol{beta}| = |boldsymbol{alpha} - boldsymbol{beta}|), we need to find the angle between them.First, we square both sides of the given equation:[|boldsymbol{alpha} + boldsymbol{beta}|^2 = |boldsymbol{alpha} - boldsymbol{beta}|^2]Expanding both sides using the dot product, we get:[(boldsymbol{alpha} + boldsymbol{beta}) cdot (boldsymbol{alpha} + boldsymbol{beta}) = (boldsymbol{alpha} - boldsymbol{beta}) cdot (boldsymbol{alpha} - boldsymbol{beta})]Simplifying both sides:[|boldsymbol{alpha}|^2 + 2boldsymbol{alpha} cdot boldsymbol{beta} + |boldsymbol{beta}|^2 = |boldsymbol{alpha}|^2 - 2boldsymbol{alpha} cdot boldsymbol{beta} + |boldsymbol{beta}|^2]Subtracting (|boldsymbol{alpha}|^2) and (|boldsymbol{beta}|^2) from both sides:[2boldsymbol{alpha} cdot boldsymbol{beta} = -2boldsymbol{alpha} cdot boldsymbol{beta}]This simplifies to:[4boldsymbol{alpha} cdot boldsymbol{beta} = 0 implies boldsymbol{alpha} cdot boldsymbol{beta} = 0]The dot product (boldsymbol{alpha} cdot boldsymbol{beta} = 0) implies that the vectors are orthogonal. Therefore, the angle (theta) between (boldsymbol{alpha}) and (boldsymbol{beta}) is (frac{pi}{2}) radians (or 90 degrees).Thus, the angle between (boldsymbol{alpha}) and (boldsymbol{beta}) is (boxed{dfrac{pi}{2}}).,<think>Okay, let me try to figure this problem out. So, we have two non-zero vectors, alpha and beta, and they satisfy the condition that the magnitude of alpha plus beta is equal to the magnitude of alpha minus beta. We need to find the angle between them. Hmm, right. Let me recall some vector properties here.First, I remember that the magnitude squared of a vector sum can be expanded using the dot product. Specifically, for any two vectors a and b, |a + b|² = |a|² + 2a·b + |b|². Similarly, |a - b|² = |a|² - 2a·b + |b|². So maybe if I use these formulas, I can set up an equation based on the given condition.Given that |α + β| = |α - β|, if I square both sides to eliminate the square roots (since magnitudes are non-negative), I should get |α + β|² = |α - β|². Then substituting the expansions from the dot product:Left side: |α|² + 2α·β + |β|²Right side: |α|² - 2α·β + |β|²So setting them equal:|α|² + 2α·β + |β|² = |α|² - 2α·β + |β|²Hmm, let's subtract |α|² and |β|² from both sides to simplify:2α·β = -2α·βAdding 2α·β to both sides:4α·β = 0Therefore, α·β = 0Wait, the dot product of α and β is zero? That means they are orthogonal! So the angle between them must be 90 degrees, or π/2 radians. Is that right?Let me double-check. If the dot product is zero, then by definition the cosine of the angle between them is zero (since α·β = |α||β|cosθ). So cosθ = 0 implies θ = π/2. Yes, that makes sense. So the angle is 90 degrees.But let me make sure there are no other possibilities. Cosine theta equals zero only at 90 degrees and 270 degrees, but angles between vectors are always considered in the range [0, π], so 270 degrees isn't considered here. So yes, the only angle between 0 and 180 degrees (or 0 and π radians) where cosine is zero is 90 degrees.Therefore, the answer should be 90 degrees or π/2 radians. The question doesn't specify the form, but in math problems like this, radians are often preferred, especially since angles in higher mathematics are typically in radians. So π/2.Wait, but just to make sure, let me re-examine the problem. It says non-zero vectors. So both vectors have non-zero magnitude, which is important because if one was zero, the dot product might not be defined. But since both are non-zero, their magnitudes are positive, so dividing by |α||β| is allowed. So cosθ = 0 is valid, leading to theta = π/2. So yeah, the angle is right angle. I think that's it.I don't see any mistakes in the reasoning. Starting from the given condition, expanding using the formula for magnitude squared, simplifying, arriving at the dot product being zero, and concluding orthogonality. Seems solid. Maybe another way to think about it: if the vectors are such that the magnitude of their sum and difference are equal, then they form a rectangle's diagonals, which are equal in length only if the parallelogram is a rectangle, hence the vectors are perpendicular. That's another way to visualize it. So that also gives the same conclusion.Yep, that's right. In a parallelogram, the sum and difference of the vectors represent the diagonals. If the diagonals are equal in length, the parallelogram is a rectangle. Therefore, the original vectors must be perpendicular. So that's another confirmation. So I think confident now that the angle is π/2.Final AnswerThe angle between boldsymbol{alpha} and boldsymbol{beta} is boxed{dfrac{pi}{2}}.</think>Given non-zero vectors (boldsymbol{alpha}) and (boldsymbol{beta}) that satisfy (|boldsymbol{alpha} + boldsymbol{beta}| = |boldsymbol{alpha} - boldsymbol{beta}|), we need to find the angle between them.First, we square both sides of the given equation to eliminate the magnitudes:[|boldsymbol{alpha} + boldsymbol{beta}|^2 = |boldsymbol{alpha} - boldsymbol{beta}|^2]Expanding both sides using the dot product properties:[|boldsymbol{alpha}|^2 + 2boldsymbol{alpha} cdot boldsymbol{beta} + |boldsymbol{beta}|^2 = |boldsymbol{alpha}|^2 - 2boldsymbol{alpha} cdot boldsymbol{beta} + |boldsymbol{beta}|^2]Subtracting (|boldsymbol{alpha}|^2) and (|boldsymbol{beta}|^2) from both sides, we get:[2boldsymbol{alpha} cdot boldsymbol{beta} = -2boldsymbol{alpha} cdot boldsymbol{beta}]Adding (2boldsymbol{alpha} cdot boldsymbol{beta}) to both sides:[4boldsymbol{alpha} cdot boldsymbol{beta} = 0]This simplifies to:[boldsymbol{alpha} cdot boldsymbol{beta} = 0]The dot product (boldsymbol{alpha} cdot boldsymbol{beta} = 0) implies that the vectors are orthogonal. Therefore, the angle (theta) between (boldsymbol{alpha}) and (boldsymbol{beta}) is (90^circ) or (frac{pi}{2}) radians.[boxed{dfrac{pi}{2}}]]

🔑:4. 90^{circ} Hint: According to the parallelogram rule for the sum and difference of vectors, |boldsymbol{alpha}+boldsymbol{beta}|,|boldsymbol{alpha}-boldsymbol{beta}| are exactly the lengths of the two diagonals of the parallelogram with boldsymbol{alpha} 、 boldsymbol{beta} as adjacent sides. Since |boldsymbol{alpha}+boldsymbol{beta}|=|boldsymbol{alpha}-boldsymbol{beta}|, the parallelogram is a rectangle. Therefore, the angle between boldsymbol{alpha} and boldsymbol{beta} is 90^{circ}.

🧠:[<think>Okay, so I need to solve this physics problem about a lead bullet falling onto a solid plane. Let me read the problem again to make sure I understand it.A lead bullet at 20°C falls freely from a height of 100 m onto a solid plane. The collision is completely inelastic, and all the "apparently lost energy" is converted into heat absorbed by the bullet. The questions are: 1. What's the temperature of the bullet immediately after the fall?2. What speed would the bullet need to be dropped from (from above) to raise its temperature to the melting point of lead?Given data: - Specific heat of lead, C = 0.0315 (units not given, but probably cal/(g°C) since it's common for specific heat)- Mechanical equivalent of heat, E = 425 (probably J/cal, to convert calories to joules)- Melting point of lead, T = 335°C- Gravitational acceleration, g = 9.80 m/s²Wait, the units for specific heat and mechanical equivalent are crucial. The mechanical equivalent of heat is the number of joules per calorie. So 425 J/cal? Wait, normally the mechanical equivalent is 4.184 J/cal, so 425 seems high. Maybe there's a unit conversion here. Let me check. Wait, perhaps the problem is using different units. Wait, specific heat C is given as 0.0315. If it's in cal/(g°C), then lead's specific heat capacity is about 0.128 J/(g°C). But 0.0315 cal/(g°C) would be 0.0315 * 4.184 ≈ 0.131 J/(g°C), which is close. But the given E is 425. Hmm, perhaps in the problem's context, E is given in some other unit. Maybe it's 425 J per kilocalorie? Wait, no. Let's hold on. Let me see.Alternatively, perhaps in this problem, the specific heat C is given in kcal/(kg°C), which would make the numbers different. Wait, the problem states "mechanical equivalent of heat E=425". The mechanical equivalent of heat is the conversion factor between mechanical energy (in joules) and heat (in calories). Normally, 1 calorie = 4.184 joules, so 1 joule ≈ 0.239 calories. So the mechanical equivalent of heat is usually 4.184 J/cal. But here it's given as 425. That seems way too high. Wait, maybe the units are different. Let me check again.Wait, the problem says: specific heat of lead C=0.0315. If that's in cal/(g°C), then for lead, specific heat is about 0.128 J/(g°C), which converts to 0.128 / 4.184 ≈ 0.0306 cal/(g°C). So 0.0315 is close, maybe rounded. So that seems plausible.Then, the mechanical equivalent E=425. Wait, if E is in J/cal, then 425 would mean 425 joules per calorie, which is way too high because 1 calorie is 4.184 joules. So maybe E is given in different units. Wait, maybe the problem is using E as the conversion from kilogram-meters to calories? Wait, old units sometimes used kilogram-meters (kgf·m) to calories. Let me think.Alternatively, maybe the problem is using E as the number of joules equivalent to 1 kcal. Since 1 kcal = 4184 J, so if E is 425, that would not make sense either. Hmm. Alternatively, maybe E is given as 425 J/(kg·°C)? No, that's specific heat. Wait, maybe there's a typo, but the user just copied the problem as given. Let me check the original problem again.Wait, the problem says: "mechanical equivalent of heat E=425". The term "mechanical equivalent of heat" historically refers to how many mechanical energy units correspond to a unit of heat. For example, Joule's experiment showed that 1 calorie is equivalent to about 4.184 joules. So if E is 425, maybe it's in different units. For instance, if energy is in some other unit, like kilogram-force meters. Wait, 1 calorie = 4.184 joules, and 1 joule = 1 kg·m²/s². If we use kilogram-force meters, 1 kgf·m = 9.8 N·m = 9.8 J. So if E is given in kgf·m per calorie, then 1 calorie = 4.184 J = 4.184 / 9.8 ≈ 0.427 kgf·m. So E would be approximately 0.427 kgf·m/cal. But the problem states E=425, which is way larger. Maybe it's in some other unit. Alternatively, perhaps the problem is using E as the number of Joules per kilocalorie? Since 1 kilocalorie = 4184 J, so E would be 4184 J/kcal. But the problem states E=425. Hmm.Alternatively, perhaps the problem is written with a different unit system. For example, if specific heat is given in cal/(kg°C) instead of cal/(g°C). Wait, if C is 0.0315 cal/(kg°C), that would be way too low, because specific heat per gram is 0.0315 cal/(g°C), so per kilogram it would be 31.5 cal/(kg°C). That seems more reasonable. Wait, but the problem states C=0.0315. If the unit is kcal/(kg°C), then 0.0315 kcal/(kg°C) is 31.5 cal/(kg°C). But lead's specific heat is around 128 J/(kg°C), which is 128 / 4184 ≈ 0.0306 cal/(g°C) or 30.6 cal/(kg°C). So 31.5 cal/(kg°C) is close, so maybe C is given in cal/(kg°C) but written as 0.0315? That seems inconsistent. Wait, 31.5 cal/(kg°C) is 0.0315 kcal/(kg°C). Maybe the problem uses kilocalories. So if C is 0.0315 kcal/(kg°C), then that's 31.5 cal/(kg°C). But then the mechanical equivalent would need to be in J per kcal. If E is 425 J/kcal, then that's 0.425 J/cal, which is the inverse of the usual. Wait, no. If 1 kcal = 425 J, then E=425 J/kcal. But 1 kcal is actually 4184 J, so this is off by a factor of 10. Hmm. This is confusing.Alternatively, maybe the problem is using grams instead of kilograms. Let me think. Let's try to proceed step by step.First, question 1: Temperature after the fall.The bullet falls from 100 m. The potential energy is converted into kinetic energy, and upon impact, since the collision is completely inelastic, all the kinetic energy is lost (assuming it stops completely). But the problem says "all the apparently lost energy was converted into heat absorbed by the bullet". So the energy lost during the collision is converted into heat. But first, the bullet is falling from 100 m, so its potential energy is mgh. When it hits the ground, it has kinetic energy (1/2)mv², which would be equal to mgh if there's no air resistance. Then, upon impact, the kinetic energy is lost, converted into heat. So the heat energy Q is equal to mgh. Then, using Q = mcΔT, where c is specific heat capacity, we can find the temperature rise.But the problem mentions the mechanical equivalent of heat E. So maybe they use the formula Q = (Energy)/E, where E is the conversion factor from mechanical energy to heat. Let me recall. In older systems, the mechanical equivalent of heat is the amount of mechanical energy needed to produce one unit of heat. For example, 4.184 J = 1 cal, so E would be 4.184 J/cal. If E is given as 425, maybe in different units. Wait, perhaps E is in J/kilocal? Then 1 kilocal = 4184 J, so E would be 4184 J/kcal. If in the problem, E=425, which is close to 4184, but off by a factor of 10. Maybe a typo? Alternatively, perhaps E is given in J/cal, and it's 4.25, but written as 425. Maybe the decimal is missing. But the problem says E=425. Hmm.Alternatively, maybe the units are in CGS system. Let's see. 1 erg = 1 g·cm²/s². 1 calorie = 4.184 J = 4.184×10^7 erg. So if E is in erg/cal, then E=4.184×10^7 erg/cal. But 425 is way smaller. Not sure.Alternatively, maybe the problem uses a different definition. Let's try to proceed with the given numbers. Let's see.So first, calculate the potential energy: mgh. But we don't know the mass. However, since we are dealing with specific heat per unit mass, perhaps the mass cancels out.Let me write the equations.The energy converted into heat is equal to the mechanical energy lost, which is mgh (assuming all potential energy is converted into kinetic energy, and then all kinetic energy is converted into heat). The heat Q absorbed by the bullet is Q = mcΔT, where m is mass, c is specific heat, ΔT is temperature change.But since the mechanical equivalent of heat E is given, perhaps the formula is Energy = Q * E, so mgh = Q * E. Then Q = mgh / E. But Q is also mcΔT. So:mcΔT = mgh / ECancel mass m:cΔT = gh / ETherefore, ΔT = gh / (E c)But wait, need to check units. Let's see.If c is in cal/(g°C), E is J/cal, g is m/s², h is meters.So, let's convert all units.First, g = 9.80 m/s²h = 100 mc = 0.0315 cal/(g°C). Let's convert to J/(kg°C). Since 1 cal = 4.184 J, so c = 0.0315 cal/(g°C) * 4.184 J/cal * 1000 g/kg = 0.0315 * 4.184 * 1000 J/(kg°C) ≈ 0.0315 * 4184 ≈ 131.7 J/(kg°C)E is given as 425. If E is J/cal, then 1 cal = 425 J? Wait, that can't be. Normally, 1 cal = 4.184 J, so E should be 4.184 J/cal. If E is 425, then this would be 425 J per calorie, which is 100 times larger. So maybe there is a mistake here.Alternatively, maybe E is given in some other unit. Wait, the problem says "mechanical equivalent of heat E=425". In old textbooks, the mechanical equivalent of heat is sometimes expressed as the amount of mechanical energy (in joules or ergs) equivalent to 1 calorie. So if E=4.184 J/cal, that's correct. But here it's 425, which is two orders of magnitude higher. Alternatively, maybe the problem is using E as 425 J/g°C? No, that doesn't make sense.Alternatively, perhaps the problem is mixing units. Let's assume that E=425 is in J/kg, but that doesn't seem to fit.Wait, perhaps the problem is in a different unit system. For example, if the specific heat is given in cal/(kg°C), then c=0.0315 cal/(kg°C). That would be very low, but if E is 425 cal/J, that also seems odd.Alternatively, maybe the problem uses E as the conversion from energy in kg·m to calories. For instance, 1 kg·m = 9.8 J = 9.8 / 4.184 ≈ 2.34 cal. So if the potential energy is mgh in kg·m, then Q = mgh / E, where E is in cal per kg·m. If E=425, that would be 425 cal per kg·m, which is way too high. Because 1 kg·m is about 2.34 cal. So this is confusing.Alternatively, perhaps the problem is written with E in a different way. Let's look at the formula.In some older texts, the heat produced is Q = (Energy) / J, where J is the mechanical equivalent of heat. So if the energy is in joules, and Q is in calories, then J = 4.184 J/cal. But if they write Q = Energy / J, then J would have units of J/cal.But in the problem, if E=425, and if we are to use Q = Energy / E, then Q would be in the same units as Energy divided by E. Let's check:If the potential energy is mgh (in joules), then Q (in calories) would be (mgh) / E. So E must have units of J/cal. If E=425 J/cal, then to get Q in cal, we divide the energy in J by E. But 425 J/cal is way too high, since 1 cal = 4.184 J. So maybe the problem actually uses E=4.25 J/cal, and there's a decimal missing? If E=4.25 J/cal, that's close to the actual 4.184. But the problem says 425. Maybe a translation error? The original problem might have commas as decimal separators. If it's written as 4,25, that would be 4.25 in some countries. So maybe E=4.25 J/cal? Let's check.Assuming E=4.25 J/cal, then specific heat c=0.0315 cal/(g°C). Let's compute ΔT.The energy from the fall is mgh. Convert that to calories: Q = mgh / E. Then Q = mcΔT. So:mgh / E = mcΔTCancel mass m:gh / E = cΔTΔT = (gh) / (cE)Plugging numbers:g=9.80 m/s², h=100 m, c=0.0315 cal/(g°C), E=4.25 J/cal.But need to make sure units are consistent.First, convert c to J/(g°C). Since 1 cal = 4.25 J (if E=4.25 J/cal), then c = 0.0315 cal/(g°C) * 4.25 J/cal = 0.0315 * 4.25 J/(g°C) ≈ 0.134 J/(g°C) = 134 J/(kg°C) (since 1 kg = 1000 g).Wait, no, actually:Wait, if c is 0.0315 cal/(g°C), and 1 cal = 4.25 J, then c = 0.0315 * 4.25 J/(g°C) ≈ 0.134 J/(g°C), which is 134 J/(kg°C). That's plausible because the specific heat capacity of lead is around 128 J/(kg°C). So this seems reasonable. So if E=4.25 J/cal, which is close to the actual 4.184, then this makes sense. So probably a comma was used instead of a decimal, making E=4,25 in the original, which translates to 4.25.So proceeding with E=4.25 J/cal.Then, ΔT = (gh) / (cE). Wait, let's see:Wait, the equation was:ΔT = (gh) / (cE)But no, let's rederive it.Energy (J) = mgh.Heat (cal) = Energy (J) / E (J/cal) = mgh / E.Heat (cal) = Q = mcΔT (where c is in cal/(g°C), m is in grams). Wait, but units need to match.Wait, m in grams. c in cal/(g°C). So Q = mcΔT (cal).But Energy (mgh) is in J. So converting Energy to cal: Q = (mgh) / E.Thus:mcΔT = (mgh)/ECancel m:cΔT = (gh)/ETherefore, ΔT = (gh) / (cE)But let's check units:g is in m/s², h in meters, c in cal/(g°C), E in J/cal.So m/s² * m = (m²/s²) (energy per unit mass?)Wait, let's do unit conversions step by step.Compute numerator: gh = 9.80 m/s² * 100 m = 980 m²/s².Now, denominator: c * E.c = 0.0315 cal/(g°C). E = 4.25 J/cal.So c * E = 0.0315 cal/(g°C) * 4.25 J/cal = 0.0315 * 4.25 J/(g°C) ≈ 0.134 J/(g°C) = 134 J/(kg°C) (since 1 kg = 1000 g). So 134 J/(kg°C).So ΔT = gh / (c * E) = (980 m²/s²) / (134 J/(kg°C))But J = kg·m²/s². So:ΔT = (980 m²/s²) / [134 (kg·m²/s²)/(kg·°C)) ] = (980 / 134) °C ≈ 7.31 °CSo the temperature rise is approximately 7.31°C. Since the initial temperature is 20°C, the final temperature is 20 + 7.31 ≈ 27.31°C. Rounded to maybe 27.3°C.But let's check the calculation again.Wait, ΔT = (gh)/(cE). Wait, but with units properly considered.Wait, gh has units of m²/s². c is in cal/(g°C). E is in J/cal. So c*E is (cal/(g°C))*(J/cal) = J/(g°C). So gh (m²/s²) divided by J/(g°C). But J = kg·m²/s². So J/(g°C) = (kg·m²/s²)/(g°C) = (1000 g·m²/s²)/(g°C) = 1000 m²/(s²·°C). Therefore:ΔT = (m²/s²) / (1000 m²/(s²·°C)) ) = (1 / 1000) °C. Wait, that can't be. Wait, no. Wait, I think the unit conversion is more involved.Wait, let's compute the units step by step:gh: 9.80 m/s² * 100 m = 980 m²/s².c = 0.0315 cal/(g°C)E = 4.25 J/calc*E = 0.0315 cal/(g°C) * 4.25 J/cal = 0.134 J/(g°C)Convert J to kg·m²/s²: 0.134 kg·m²/(s²·g°C) (since 1 J = 1 kg·m²/s²)But 1 kg = 1000 g, so:0.134 kg·m²/(s²·g°C) = 0.134 * 1000 g·m²/(s²·g°C) = 134 m²/(s²·°C)Therefore, ΔT = (980 m²/s²) / (134 m²/(s²·°C)) ) = 980 / 134 °C ≈ 7.31°CYes, so ΔT ≈7.31°C. Therefore, final temperature is 20 + 7.31 ≈27.3°C.So answer to question 1 is approximately 27.3°C.But let me check again with E=425. If E was actually 425, how would that work? If E=425 J/cal, which is 100 times larger than normal.Then, c*E = 0.0315 cal/(g°C)*425 J/cal = 13.3875 J/(g°C) = 13387.5 J/(kg°C)Then, ΔT = (980 m²/s²) / (13387.5 m²/(s²·°C)) ) ≈ 0.0732°C. So temp rise is 0.073°C, which would make the final temp 20.073°C. That seems too low. So clearly, E=425 is not J/cal. Therefore, the problem must have a typo, and E=4.25 is the intended value. So I think the correct approach is with E=4.25 J/cal.Therefore, the answer to part 1 is approximately 27.3°C.For part 2: What speed would the bullet need to be dropped from to raise its temperature to the melting point of lead (335°C). So the temperature rise needed is 335 - 20 = 315°C.Using the same approach: ΔT = (v²/(2)) / (c*E). Wait, when you drop something, the kinetic energy is (1/2)mv². If it's dropped from a height h, then v = sqrt(2gh). But if we need to find the speed required to generate enough heat to raise the temperature by 315°C, then:The energy converted to heat is (1/2)mv² (if it's stopped completely). Then, Q = (1/2)mv² / E = mcΔTCancel m:(1/2)v² / E = cΔTTherefore, v² = 2 E c ΔTSo v = sqrt(2 E c ΔT)Again, using E=4.25 J/cal, c=0.0315 cal/(g°C), ΔT=315°C.First, compute 2*E*c*ΔT:2 * 4.25 J/cal * 0.0315 cal/(g°C) * 315°CUnits: J/cal * cal/(g°C) * °C = J/gSo 2*4.25*0.0315*315 J/gCalculate the numerical value:First, compute 4.25 * 0.0315 = 0.133875Then, 0.133875 * 315 = 42.190625Then, 2 * 42.190625 = 84.38125 J/gConvert J/g to m²/s² (since 1 J = 1 kg·m²/s² and 1 g = 0.001 kg):84.38125 J/g = 84.38125 (kg·m²/s²)/0.001 kg = 84.38125 * 1000 m²/s² = 84381.25 m²/s²Therefore, v = sqrt(84381.25) ≈ 290.48 m/sSo the required speed is approximately 290.5 m/s.But let's check the calculation step by step:2 * 4.25 = 8.58.5 * 0.0315 = 0.267750.26775 * 315 = 84.38125So yes, 84.38125 J/gSince 1 J/g = 1000 J/kg, and 1 J/kg = 1 m²/s², so 84.38125 J/g = 84381.25 J/kg = 84381.25 m²/s²Therefore, v = sqrt(84381.25) ≈ 290.48 m/sAlternatively, if we use E=425 J/cal (which is probably wrong), we would get:v² = 2 * 425 * 0.0315 * 315But that would be way too large. Let's see:2*425=850850*0.0315=26.77526.775*315=8431.125v=sqrt(8431.125*1000) = sqrt(8,431,125) ≈ 2903 m/s, which is way too high for a bullet drop speed. So clearly, E=4.25 is the correct value here.Alternatively, maybe the speed is calculated from the height. Since if you need v=290.5 m/s, then the height would be h = v²/(2g) = (290.5)^2/(2*9.8) ≈ (84380)/(19.6) ≈ 4305 m. So the bullet would need to be dropped from ~4305 m to reach that speed, but if we have air resistance, the actual speed would be limited. However, the problem states "all the apparently lost energy was converted into heat", so maybe they are neglecting air resistance, so the height needed would be h = v²/(2g). But the question asks for speed, not height. So the answer is 290.5 m/s.But let's check the calculation again.Alternatively, using the same formula as before, ΔT = (KE) / (m c E)Wait, no, KE = (1/2)mv², so Q = KE / E = (1/2 mv²)/E. Then Q = mcΔT, so:(1/2 mv²)/E = mcΔT => v² / (2E) = cΔT => v = sqrt(2 E c ΔT )Which is what we did.So with E=4.25 J/cal, c=0.0315 cal/(g°C), ΔT=315°C:v = sqrt(2 * 4.25 * 0.0315 * 315 * 10^3) ??? Wait, no, wait. Let's check units again.Wait, E is in J/cal. c is in cal/(g°C). So 2*E*c*ΔT is 2*(J/cal)*(cal/(g°C))*°C = 2*J/(g°C)*°C = 2*J/g. Therefore, v² = 2*E*c*ΔT = 2*J/g. Since 1 J = 1 kg·m²/s², and 1 g = 0.001 kg, so J/g = (kg·m²/s²)/g = (kg·m²/s²)/(0.001 kg) = 1000 m²/s². Therefore, 2*E*c*ΔT has units of 1000 m²/s². Wait, no:Wait, 2*E*c*ΔT:E [J/cal] * c [cal/(g°C)] * ΔT [°C] = (J/cal) * (cal/(g°C)) * °C = J/(g°C) * °C = J/g. Then, multiply by 2: 2*J/g. Then, v² = 2*E*c*ΔT [J/g] = (kg·m²/s²)/g = (1000 g·m²/s²)/g = 1000 m²/s². Therefore, v² = 1000 m²/s² * numerical value.Wait, no. Let me compute 2*E*c*ΔT:If E=4.25 J/cal, c=0.0315 cal/(g°C), ΔT=315°C.So 2*4.25 J/cal * 0.0315 cal/(g°C) * 315°CThe cals cancel, °C cancels, so:2*4.25*0.0315*315 J/gCompute this:2*4.25 = 8.58.5 * 0.0315 = 0.267750.26775 * 315 = 84.38125So 84.38125 J/gBut J/g = 1000 J/kg. So 84.38125 J/g = 84381.25 J/kgSince 1 J/kg = 1 m²/s², so v² = 84381.25 m²/s²Therefore, v = sqrt(84381.25) ≈ 290.48 m/sYes, same result. So approximately 290.5 m/s.Therefore, the bullet would need to be dropped with a speed of approximately 290.5 m/s. But since the problem asks "What speed would the bullet need to be dropped from above", maybe they are asking for the equivalent height, but no, the question says "what speed", so it's the impact speed required. But if it's dropped from a height h, then the speed would be sqrt(2gh). So if they need the speed to be 290.5 m/s, then the height would be h = v²/(2g) ≈ (290.5)^2 / (2*9.8) ≈ 84380 / 19.6 ≈ 4305 m. But the question specifically asks for the speed, not the height. So answer is 290.5 m/s.But let's verify with E=4.25 J/cal.Alternatively, let's think in terms of energy:Potential energy per unit mass: gh = 9.8*100 = 980 J/kg (if E=1 J=1 J). But with E=4.25 J/cal, so potential energy per unit mass is mgh = m*9.8*100 = 980m J. Then, heat Q = 980m / 4.25 cal. Then, Q = 980/(4.25) cal/kg ≈ 230.588 cal/kg.Specific heat c=0.0315 cal/(g°C) = 31.5 cal/(kg°C). So ΔT = Q / c = 230.588 / 31.5 ≈7.31°C. Which matches the previous result. So yes, with E=4.25 J/cal, calculations are consistent.Thus, the answers are:1. Approximately 27.3°C.2. Approximately 290.5 m/s.But let me check significant figures. The given data:C=0.0315 (4 significant figures)E=425 (3 significant figures, unless it's 4.25e2, but written as 425, maybe 3)g=9.80 (3 significant figures)h=100 (1 or 3? If 100 is exact, but probably 3: 1.00e2)Melting point T=335 (3 significant figures)So the calculations:For part 1:ΔT=7.31... which is 7.3°C (2 decimal points, but given the significant figures in E=425 (if 3), then 7.31≈7.3°C. So final temp 20 +7.3=27.3°C.For part 2:v≈290.5 m/s. Given the inputs, maybe round to 290 m/s or 291 m/s. But 290.5 is precise, but since E=425 is maybe 3 significant figures, so 291 m/s.But since in the problem statement E=425, which if taken as 3 significant figures, then the answer should have 3. So 290 m/s (if 425 is 3) or 291. But the exact value is ~290.5.But let me check with E=425, if I was wrong before. Wait, but we established that E=425 is likely a typo, and should be 4.25. However, if the problem insists on E=425, then the answers would be as follows:For part 1:ΔT = (9.80*100)/(0.0315*425) ≈980/(13.3875)≈73.2°C. Then final temp 20+73.2=93.2°C. Which seems high, but possible.For part 2:v= sqrt(2*425*0.0315*315)≈sqrt(2*425*0.0315*315)=sqrt(2*425*9.9225)=sqrt(2*4217.0625)=sqrt(8434.125)=91.84 m/s. Wait, no, wait:Wait, 425*0.0315=13.387513.3875*315=4217.06252*4217.0625=8434.125sqrt(8434.125)=91.84 m/s.But that contradicts the previous if E=425.So, depending on the value of E, the answer changes dramatically. But since with E=4.25 we get reasonable answers, matching the specific heat of lead, and with E=425, the answers are either too low or inconsistent with known physics, I think E=4.25 is intended.Therefore, the problem likely has a typo, and E=4.25 J/cal is the correct value, corresponding to the standard mechanical equivalent of heat (~4.184 J/cal). Therefore, the answers are:1. 27.3°C2. 290.5 m/sRounded to appropriate decimal places, maybe 27°C and 291 m/s.But given the problem provides E=425, which is two orders higher than normal, I need to check if there is another interpretation.Wait, maybe the units of specific heat are different. If specific heat is given in C=0.0315 kcal/(kg°C), then 0.0315 kcal/(kg°C) = 31.5 cal/(kg°C). Then, with E=425 J/cal (which is incorrect, but let's proceed), then calculations would be:But even then, converting units would be messy. Alternatively, if the problem uses E=425 kg·m/cal (a different unit), then:1 cal = 425 kg·m. Then, potential energy mgh is in kg·m·9.8 m/s²? No, not quite. Alternatively, in some systems, work is measured in kilogram-force meters. 1 kgf·m = 9.8 J. So if E=425 kgf·m/cal, then 1 cal = 425 kgf·m = 425*9.8 J ≈4165 J, which is roughly 1 kcal (since 1 kcal=4184 J). So if E=425 kgf·m/kilocal, then 1 kcal = 425 kgf·m, which is 4165 J. Then, E=425 kgf·m/kcal = 4165 J/kcal. So 1 kcal = 4165 J, which is close to the actual 4184 J.So if C is given in kcal/(kg°C), then specific heat C=0.0315 kcal/(kg°C). Then, the calculation would be:Energy from fall: mgh = m*9.8*100 = 980m J.Convert to kcal: 980m J / 4165 J/kcal ≈0.2353m kcal.Heat required to raise temperature: Q = m*C*ΔT = m*0.0315 kcal/(kg°C)*ΔT.Set equal: 0.2353m = 0.0315m ΔT => ΔT = 0.2353 /0.0315 ≈7.47°C.So temperature rise ≈7.5°C, final temp≈27.5°C.Similarly, for part 2:Need ΔT=315°C.Q=0.0315m*315=9.9225m kcal.Energy required: 9.9225m kcal =9.9225m *4165 J/kcal≈41325m J.Kinetic energy: (1/2)mv²=41325m J => v²=82650 =>v≈287.5 m/s.So with this interpretation, E=425 kgf·m/kcal=4165 J/kcal, C=0.0315 kcal/(kg°C).This gives similar results to the previous method. So temp rise≈7.5°C, v≈287.5 m/s.This suggests that the problem might be using E=425 kgf·m/kcal, and specific heat in kcal/(kg°C). Which is plausible, given the era of the problem. So if that's the case, then:For part 1:Q = mgh / E = (m*9.8*100 kg·m) / (425 kgf·m/kcal) ). Wait, 1 kgf·m = 9.8 J, so mgh in J is m*9.8*100. To convert to kgf·m: (m*9.8*100)/9.8=100m kgf·m.Then, Q = 100m kgf·m / 425 kgf·m/kcal = (100/425) kcal ≈0.2353 kcal.Heat required: Q = m*C*ΔT = m*0.0315 kcal/(kg°C)*ΔT.Set equal:0.2353 =0.0315*ΔT =>ΔT≈7.47°C.So final temp≈27.5°C.For part 2:Need Q = m*0.0315*315≈9.9225m kcal.Energy required=9.9225m kcal *425 kgf·m/kcal ≈4217.06m kgf·m.Kinetic energy in kgf·m: (1/2)mv² /9.8 =4217.06m.So:(1/2)mv² /9.8=4217.06m =>v²=4217.06*2*9.8≈82650 =>v≈287.5 m/s.So this approach also gives 287.5 m/s.Therefore, depending on the interpretation of units, the answer is either approximately 290 m/s or 287.5 m/s. The slight difference is due to the exact value of E. Since the problem gives E=425, if we consider it as kgf·m/kcal, then the answer is approximately 287.5 m/s.However, given that the problem mentions "mechanical equivalent of heat E=425", and in standard terms, the mechanical equivalent is 4.184 J/cal or 4184 J/kcal, which is equivalent to 426.9 kgf·m/kcal (since 1 kcal=4184 J=4184/9.8≈426.9 kgf·m). So if they approximate it to 425 kgf·m/kcal, then the calculation gives 287.5 m/s. Close to the other method.Therefore, given the problem's data, the correct answers are approximately:1. 27.5°C2. 287.5 m/sBut the question says "speed would the bullet need to be dropped from above", which is a bit ambiguous. If it's asking for the speed upon impact, it's 287.5 m/s. If asking for the height, h=v²/(2g)= (287.5)^2/(2*9.8)≈(82656)/19.6≈4218 m. But the question asks for speed, so answer is approximately 287.5 m/s.But given that using E=425 as kgf·m/kcal leads to 287.5 m/s, which is close to our initial calculation with E=4.25 J/cal (290.5 m/s). The difference is due to the exact value of E.Given that, and to match the problem's given numbers, it's better to present the answer as per the problem's units. Since the problem states E=425, which is likely in kgf·m/kcal, so:1. Temperature rise ΔT=7.47°C, final temp≈27.5°C2. Speed≈287.5 m/sBut since the problem might expect using E=425 J/cal (which would be incorrect but following instructions), but that leads to very low temperature rise and extremely high speed, which contradicts physics.Alternatively, if the problem uses E=425 J/kg°C, but that doesn't fit. Given the confusion, and the fact that 425 is close to 4184 J/kcal when converted via kgf·m, which is 425 kgf·m/kcal≈4165 J/kcal, which is≈1 kcal, then it's possible that the problem uses kilocalories.Therefore, given the problem's data, and considering E=425 kgf·m/kcal, then:Question 1 Answer: ~27.5°CQuestion 2 Answer: ~287.5 m/sBut the initial calculation with E=4.25 J/cal gives similar results. Therefore, it's best to present both methods, but since the problem gives E=425, probably expecting the answer using kgf·m/kcal.However, without more context on the units, it's challenging, but given the numbers, I think the problem expects the following:Given E=425 (assumed J/cal, which is incorrect but proceeding):But this gives unreasonable results. Therefore, the correct approach is assuming E=4.25 J/cal, leading to:1. 27.3°C2. 290.5 m/sBut since the problem states E=425, perhaps there's a missing decimal. If E=4.25 is correct, then answers are approximately 27.3°C and 290.5 m/s.Given that, and to match the problem's instruction to use given data, even if there's a possible typo, but assuming the data is correct as given, perhaps in a different unit system, the answer is:1. The temperature of the bullet immediately after the fall is boxed{27^circtext{C}} (approximately).2. The speed needed is boxed{290 text{m/s}} (approximately).But considering precise calculation with E=4.25 J/cal:1. 27.3°C, which rounds to 27°C.2. 290.5 m/s, rounds to 290 m/s or 291 m/s.But the problem might expect more precise answers.Alternatively, if we take E=425 as intended and use cal to J conversion, but considering the bullet's mass cancels out, the correct formula is ΔT = (gh)/(c*E), but with units properly converted.Wait, trying one last approach.Assume:- c = 0.0315 cal/(g°C)- E = 425 J/cal (even though this is too high)Compute ΔT = (gh)/(c*E)First, convert c to J/(g°C):0.0315 cal/(g°C) * 425 J/cal = 13.3875 J/(g°C) = 13387.5 J/(kg°C)Then:ΔT = (9.80 m/s² * 100 m) / (13387.5 J/(kg°C))But J = kg·m²/s², so:ΔT = (980 m²/s²) / (13387.5 m²/(s²·°C)) = 980 /13387.5 ≈0.0732°CFinal temp: 20.073°C. This is negligible, so doesn't make sense.Therefore, this can't be the right approach. Therefore, the problem must have a typo in E. Thus, the correct approach is with E=4.25 J/cal.Therefore, final answers:1. boxed{27^circtext{C}} 2. boxed{290 text{m/s}}But to be precise, 27.3°C and 290.5 m/s, but the problem might expect rounding.Final Answer1. The temperature of the bullet immediately after the fall is boxed{27^circtext{C}}.2. The speed needed to raise the bullet's temperature to the melting point of lead is boxed{290 text{m/s}}.</think>To solve the problem, we need to determine the temperature rise of the lead bullet after falling from 100 m and the speed required to raise its temperature to the melting point of lead. Key Steps:1. Temperature After the Fall: - The potential energy (PE) of the bullet is converted into heat energy. - Use the formula for temperature rise: [ Delta T = frac{gh}{cE} ] - Given: - ( g = 9.80 , text{m/s}^2 ) - ( h = 100 , text{m} ) - ( c = 0.0315 , text{cal/(g°C)} ) - ( E = 4.25 , text{J/cal} ) (assuming a typo in the problem statement, corrected from 425 to 4.25 J/cal for realistic results) - Calculations: [ Delta T = frac{9.80 times 100}{0.0315 times 4.25} approx 7.31 , text{°C} ] - Final temperature: [ 20 , text{°C} + 7.31 , text{°C} approx 27.3 , text{°C} ]2. Speed to Reach Melting Point: - The temperature rise needed is ( 335 , text{°C} - 20 , text{°C} = 315 , text{°C} ). - Use the formula for speed: [ v = sqrt{2EcDelta T} ] - Calculations: [ v = sqrt{2 times 4.25 times 0.0315 times 315 times 1000} approx 290.5 , text{m/s} ] Final Answers:1. The temperature of the bullet immediately after the fall is (boxed{27^circtext{C}}).2. The speed needed to raise the bullet's temperature to the melting point of lead is (boxed{290 text{m/s}}).,<think>Okay, so I need to solve these two physics problems about a lead bullet. Let's see. The first question is asking for the temperature of the bullet immediately after it falls 100 meters onto a solid plane. All the lost energy is converted into heat absorbed by the bullet. The second question is asking for the speed needed if the bullet is dropped from a height to raise its temperature to the melting point of lead. First, let's tackle question 1. The bullet is at 20 degrees Celsius initially, falls from 100 meters, and the impact is completely inelastic. So when it hits the ground, all the energy it had (kinetic?) is converted into heat. Wait, but since it's falling freely, its initial potential energy converts into kinetic energy as it falls. Then, upon impact, that kinetic energy is lost and converted into heat. Since it's completely inelastic, maybe all the kinetic energy goes into heat? Wait, but if it's falling freely, then the velocity just before impact can be found using kinematic equations or energy conservation. So the potential energy at height h is mgh, which converts to kinetic energy (1/2)mv². But since it's falling freely, maybe air resistance is negligible? The problem doesn't mention air resistance, so I can ignore it. So all the potential energy converts into kinetic energy, and then upon impact, all that kinetic energy is converted into heat. But wait, the problem states "all the apparently lost energy was converted into heat absorbed by the bullet". So maybe the "lost energy" here refers to the kinetic energy lost during the inelastic collision with the plane. Since it's completely inelastic, perhaps the bullet doesn't bounce back at all, so all the kinetic energy is lost. So the heat generated is equal to the initial potential energy? Wait, or is it the kinetic energy just before impact?Wait, the bullet falls from 100m, so the potential energy is mgh. If all that energy is converted into heat, then the heat Q would be mgh. Then the temperature rise would be Q = mcΔT, where c is the specific heat capacity. But here, the mechanical equivalent of heat E is given as 425. Hmm. So maybe we need to convert mechanical energy (in joules?) to calories or something? Wait, the mechanical equivalent of heat is the number of work units equivalent to one heat unit. For example, 1 calorie = 4.184 joules, so E here is 425. Maybe the units here are different. Let's check the units given.The specific heat of lead is given as C=0.0315. Hmm, units? Typically, specific heat capacity is in J/(kg·°C) or cal/(g·°C). The problem doesn't specify, but since they mention the mechanical equivalent of heat E=425, which might be converting between work (joules) and calories. Wait, maybe in this problem, the units are in some non-SI system. For example, if E is 425, which could mean 425 J corresponds to 1 calorie or something. Wait, maybe the mechanical equivalent of heat here is given in units that require conversion. Let me think.In the problem, the specific heat C is given as 0.0315. If we consider the units, in SI units, specific heat capacity of lead is around 128 J/(kg·°C). But 0.0315 might be in cal/(g·°C), which would be 0.0315 cal/(g·°C). To convert to J/(kg·°C), multiply by 4184 J/cal and 1000 g/kg, so 0.0315 * 4184 * 1000 = that's a huge number, which is way more than 128. Hmm, maybe not. Alternatively, perhaps the specific heat is given in some other unit. Wait, maybe in this problem, they are using a different system where the specific heat is given in, say, kcal/(kg·°C), and E is the conversion between mechanical work and heat.Alternatively, perhaps the units here are as follows: the mechanical equivalent of heat E is 425, which is the number of units of mechanical energy (maybe in J) equivalent to 1 unit of heat (maybe in cal). Wait, but if E is 425, like 425 J = 1 kcal or something. But I need to be careful here.Alternatively, the problem might be using the concept that heat produced Q is equal to the mechanical energy lost divided by E. Because in some older systems, the mechanical equivalent of heat is the amount of work needed to produce a unit of heat. For example, in the past, it was determined that 4.184 J = 1 cal, so the mechanical equivalent is 4.184 J/cal. But here, they give E=425, which might be in different units. Let me check the problem statement again.The problem says: "the mechanical equivalent of heat E=425". So if E is 425, then maybe the relation is Q = Energy / E. So if the energy lost is in joules, then Q would be in some heat unit, and then we can compute the temperature rise using specific heat.Wait, the specific heat of lead C is given as 0.0315. If E is 425, then perhaps Q (heat) = mechanical energy (work) / E, and then Q = m*C*ΔT. So putting it all together:mgh / E = m*C*ΔTThen, cancelling mass:gh / E = C*ΔTSo ΔT = (gh)/(E*C)Then, substituting the given values:g = 9.80 m/s²h = 100 mE = 425C = 0.0315So ΔT = (9.80 * 100) / (425 * 0.0315)Calculate numerator: 9.8*100 = 980Denominator: 425 * 0.0315 ≈ 425 * 0.03 = 12.75, 425 * 0.0015 = 0.6375; total ≈ 13.3875So ΔT ≈ 980 / 13.3875 ≈ 73.2 degrees CelsiusTherefore, the temperature rise is approximately 73.2°C. Since the initial temperature is 20°C, the final temperature would be 20 + 73.2 ≈ 93.2°C. So about 93°C. But let me check the exact calculation.Wait, 425 * 0.0315:First, 400 * 0.0315 = 12.625 * 0.0315 = 0.7875Total: 12.6 + 0.7875 = 13.3875Yes. So 980 /13.3875 ≈ 73.2. So ΔT ≈73.2. Then final temp is 20 + 73.2=93.2°C. So answer to question 1 is approximately 93.2°C. But let's do it more accurately.Alternatively, use calculator steps:980 divided by 13.3875.13.3875 * 70 = 937.12513.3875 *73= 13.3875*70=937.125 +13.3875*3=40.1625 → 937.125 +40.1625=977.2875That's 977.2875, which is close to 980. The difference is 980 -977.2875=2.7125.2.7125 /13.3875 ≈0.2026. So total is 73 +0.2026≈73.2. So same result.So ΔT≈73.2°C, final temp≈93.2°C. So answer to question 1 is 93.2°C, but maybe they want more precise. Let's check.Alternatively, maybe I made a wrong assumption. Let me confirm.The key equation is that the gravitational potential energy is converted into heat, but considering the mechanical equivalent of heat.If the mechanical work (energy) is W = mgh, then the heat Q produced is Q = W / E, where E is the mechanical equivalent. Then, the heat Q is used to raise the temperature of the bullet: Q = m*C*ΔT.Therefore, mgh / E = m*C*ΔT ⇒ gh / E = C*ΔT ⇒ ΔT = gh / (E*C)Yes, this is correct. So substituting:ΔT = (9.80 m/s² * 100 m) / (425 * 0.0315)Calculate numerator: 9.8 * 100 = 980 (m²/s²)Denominator: 425 * 0.0315 ≈13.3875 (unit?)Wait, units: The mechanical equivalent of heat E is given as 425, but units? If E is in J/cal, then mgh in J, Q in cal. Then specific heat C would be in cal/(g°C) or cal/(kg°C). Wait, but the problem states "the specific heat of lead C=0.0315". If E is 425 J/cal, then to have units matching:[gh] = (m/s² * m) = m²/s² = J/kg (since 1 J = 1 kg·m²/s², so J/kg is m²/s²)So gh is in J/kg.Then, Q = (J/kg) / (E) where E is in J/cal → (J/kg) / (J/cal) )= cal/kg.So Q (heat per unit mass) is in cal/kg.Then, specific heat C is given in cal/(kg·°C), so:ΔT = Q / C = (cal/kg) / (cal/(kg·°C)) )= °C.Therefore, if C is in cal/(kg·°C), then ΔT = gh / (E*C)So with E in J/cal, C in cal/(kg·°C), then the units work out.Therefore, ΔT = (9.80 * 100) / (425 * 0.0315) ≈ 73.2°CSo final temperature is 20 +73.2=93.2°C. So question 1 answer is approximately 93.2°C. Let's round to three significant figures: 93.2°C. The given data: g=9.80 (three sig figs), h=100 (two or three? 100 could be two or three, but written as 100 m, maybe two), E=425 (three), C=0.0315 (three). So maybe 73.2 is three sig figs, so the answer is 93.2°C. But perhaps the problem expects an integer, but not sure.Alternatively, maybe the original calculation leads to 73.2, so 93.2. But let's check if my interpretation is correct. The key is that the mechanical equivalent of heat is converting the mechanical energy into heat. So if E=425, that could mean 425 J = 1 kcal or something else. But in the absence of units, it's a bit ambiguous, but given the numbers, the above calculation seems plausible.Alternatively, if E is in the older unit, like kilogram-force meters per calorie. For example, in some contexts, the mechanical equivalent is 427 kgf·m/kcal, which is approximately 4184 J/kcal. But if E=425 kgf·m/kcal, then to use that, we need to convert the potential energy mgh in kgf·m to kcal.Wait, if the bullet falls 100 m, then the work done is mgh. If m is in kg, g in m/s², then work is in J. But if E is in kgf·m per calorie, then:Wait, 1 kgf·m = 9.8 J. So if E=425 kgf·m/cal, then 425 kgf·m = 425 *9.8 J ≈4165 J ≈1 kcal (since 1 kcal≈4184 J). So 425 kgf·m per kcal.In that case, the heat generated Q = (mgh) / E, where E is 425 kgf·m/kcal. Then, Q would be in kcal. Then, using specific heat C in kcal/(kg·°C). But in the problem, specific heat is given as 0.0315. If C is in kcal/(kg·°C), then:Q = mgh / E = (m *9.8*100) / (425 *9.8) )= (m*980)/4165≈ (m*0.235). Wait, but since the m cancels out in the temperature equation:Wait, let's start over. If E is 425 kgf·m/kcal, then the work done by gravity is m (kg) * g (m/s²) * h (m) = energy in J. But to convert to kgf·m, since 1 kgf = 9.8 N, so 1 kgf·m = 9.8 J. Therefore, work in kgf·m is (mgh)/9.8.Then, heat Q in kcal is [(mgh)/9.8] / E = (mgh)/(9.8*E)Then, Q = (mgh)/(9.8*425) kcalThen, the heat required to raise temperature is Q = m*C*ΔT, where C is in kcal/(kg·°C). Therefore:m*C*ΔT = (mgh)/(9.8*425)Cancel m:C*ΔT = (gh)/(9.8*425)Plug in numbers:C=0.0315 kcal/(kg·°C)g=9.8 m/s²h=100 mSo ΔT = (9.8*100)/(9.8*425*0.0315) )= (100)/(425*0.0315) ≈100/(13.3875)≈7.47°CWait, that gives a much lower temperature rise. But this contradicts the previous result. So which is correct?This is confusing. The problem is in the units of E and C. Since the problem statement doesn't specify units for E and C, but gives numerical values, we need to infer based on standard conventions or the given numbers.Given that E=425 and C=0.0315, and in the calculation above, if E is in J/cal, then the temperature rise is ~73°C, which seems plausible. But if E is in kgf·m/kcal, then the temperature rise is ~7.5°C, which seems low, but maybe?But the specific heat of lead is around 0.13 J/(g·°C), which is 130 J/(kg·°C). If we convert 0.0315 in some unit, let's see:If C is in kcal/(kg·°C), then 0.0315 kcal/(kg·°C) = 0.0315 *4184 J/(kg·°C) ≈131.7 J/(kg·°C), which matches the standard value. So that makes sense. So C=0.0315 kcal/(kg·°C). Then, E=425 kgf·m/kcal. Then, as per the second calculation, ΔT=7.47°C. So temperature rise is ~7.5°C, leading to 20+7.5=27.5°C. But that seems way too low, as a bullet falling 100m would have significant energy.Alternatively, if the problem uses E as the conversion from J to calories, say E=4.184 J/cal, but here E=425, which is different. Maybe in the problem's units, E=425 is in some other unit.Alternatively, maybe the problem is in a system where heat is measured in "heat units" where 1 heat unit = 425 J. Then, Q = mgh / 425, and then Q = m*C*ΔT. Then:mgh /425 = m*C*ΔT ⇒ ΔT = gh/(425*C) = (9.8*100)/(425*0.0315)≈ same as before, 73.2°C. So that's the same as the first calculation. Therefore, the temperature rise is 73.2°C, final temperature 93.2°C.Given that specific heat of lead is around 130 J/(kg·°C), which is 0.13 kJ/(kg·°C). If in the problem, C is given as 0.0315, perhaps in kJ/(kg·°C), then 0.0315 kJ/(kg·°C)=31.5 J/(kg·°C), which is way too low. So that can't be. Alternatively, C=0.0315 in cal/(g·°C). 1 cal/(g·°C)=4184 J/(kg·°C). So 0.0315 cal/(g·°C)=0.0315*4184≈131.7 J/(kg·°C), which is correct. Therefore, if C is 0.0315 cal/(g·°C), and E is the mechanical equivalent of heat, 425, which would be the number of joules per calorie. Wait, 1 cal=4.184 J, so if E=4.184 J/cal, but here E=425, which is different. Wait, maybe E is in some other units. For example, if the problem uses E=425 J/cal, which is not correct, but if we take it as given, then the calculation would be:Q = mgh / E = (m*9.8*100)/425 = (980m)/425. Then Q in cal. Then Q = m*C*ΔT ⇒ (980m)/425 = m*0.0315*ΔT ⇒ ΔT=980/(425*0.0315)= same as before,≈73.2°C. So this suggests that the answer is 93.2°C.Alternatively, maybe the problem is mixing units. For example, if the specific heat is given in cal/(g·°C), which is 0.0315 cal/(g·°C) =31.5 cal/(kg·°C). Then, the calculation would be:Heat required Q = m*C*ΔT = m*31.5*ΔT calMechanical energy W = mgh = m*9.8*100 JConvert W to calories: W/E = (m*980)/425 calSet equal: (980m)/425 =31.5 m ΔT ⇒ ΔT=980/(425*31.5)≈980/(13387.5)=≈0.0732°C. That's way too low, so that can't be.Hence, probably the correct interpretation is that E=425 is the number of mechanical energy units (in J) per heat unit (in cal). So 425 J/cal. But that would make 1 cal=425 J, which is not correct (since 1 cal=4.184 J). But since the problem gives E=425, we have to go with that.Alternatively, maybe the problem is set in a non-standard unit system where, for example, the mechanical equivalent is 425, meaning 425 units of work correspond to 1 unit of heat. Without knowing the actual units, it's ambiguous. However, given the numbers, and the fact that the specific heat of lead in cal/(g·°C) is about 0.031, which matches the given C=0.0315, then the correct approach is:Heat generated Q = mgh / EWhere E=425 J/cal. Then Q is in cal.Then, the heat required to raise the temperature is Q = m*C*ΔT, with C in cal/(g·°C). Wait, but units need to be consistent. If Q is in cal, m in grams, C in cal/(g·°C), then:m (grams) * C (cal/(g·°C)) * ΔT (°C) = Q (cal)But the potential energy is m (kg) * g h. Wait, no. If the bullet's mass is m kg, then mgh is in J. Then Q = mgh / E (cal), where E is in J/cal.But if the mass is in kg, then to use specific heat in cal/(g·°C), we need to convert mass to grams. So m (kg) = 1000 grams. Wait, this is getting complicated. Let's think again.Assume the bullet's mass is m kg. The potential energy is mgh J. Convert that to calories: Q = (mgh)/E cal, where E=425 J/cal. Then, the heat required to raise the temperature is Q = m * C * ΔT, but here, C is given in cal/(g·°C). Wait, so if C is cal/(g·°C), and the mass m is in kg, we need to convert to grams:m kg = 1000 m gramsTherefore, Q = (1000 m grams) * C (cal/(g·°C)) * ΔT = 1000 m * C * ΔT calSet equal to Q from energy:(mgh)/E =1000 m C ΔTCancel m:gh/E =1000 C ΔTThus,ΔT = gh/(1000 E C)Plugging in numbers:g=9.8, h=100, E=425, C=0.0315ΔT= (9.8*100)/(1000*425*0.0315)=980/(13387.5)≈0.0732°CThat's 0.073°C, which is way too low, clearly not correct. So this suggests that my previous assumption was wrong.Alternatively, if the specific heat is given in cal/(kg·°C), which is unusual, then:Q = m * C * ΔT, with m in kg, C in cal/(kg·°C), so Q in cal.But then mgh/E (where E=425 J/cal) is in cal.Thus:mgh/E = m*C*ΔT ⇒ gh/E = C*ΔT ⇒ΔT=gh/(E*C)= same as first calculation, (9.8*100)/(425*0.0315)=73.2°C. So temperature rise 73.2°C, final temp 93.2°C.But if C is in cal/(kg·°C), then 0.0315 cal/(kg·°C) is 0.0315 cal/(kg·°C)=0.0315*1000 cal/(tonne·°C). But specific heat is usually per gram or per kilogram. But 0.0315 cal/(kg·°C) is equivalent to 0.0000315 cal/(g·°C), which is way too small. Because the specific heat of lead is about 0.031 cal/(g·°C). So if C is given as 0.0315 cal/(g·°C), which is 31.5 cal/(kg·°C), but in the problem it's written as 0.0315, maybe they missed a unit prefix. This is very confusing.Alternatively, perhaps the units are in a system where E=425 is unitless, and all the energy is converted directly. But since the problem gives specific heat C=0.0315 without units, perhaps it's in some specific system.Alternatively, the problem is using CGS units. In CGS, g is 980 cm/s², which is 9.8 m/s². The mechanical equivalent of heat in CGS is 4.184×10^7 erg/cal (since 1 J=10^7 erg, and 1 cal=4.184 J). But given that E=425, maybe in this problem, it's simplified to E=425 in some unit.But this is getting too convoluted. Given the information provided, and the numbers, the most plausible approach is the first one: assuming that the potential energy mgh is converted into heat Q= mgh/E, where E is the mechanical equivalent (work per heat unit), and then Q= m*C*ΔT, leading to ΔT= gh/(E*C). Plugging in the numbers gives 73.2°C temperature rise, leading to 93.2°C final temperature. Therefore, answer 1 is 93.2°C.For question 2, what speed would the bullet need to be dropped from to raise its temperature to the melting point of lead, which is 335°C. So the temperature rise needed is 335 -20=315°C. So we need to find the height h such that the energy from falling converts into enough heat to raise the temperature by 315°C. Using the same equation:ΔT= gh/(E*C) ⇒ h= ΔT*E*C /gPlugging in ΔT=315°C:h=315*425*0.0315 /9.80Calculate numerator: 315*425=133,875; 133,875*0.0315≈133,875*0.03=4,016.25; 133,875*0.0015≈200.8125; total≈4,016.25+200.8125≈4,217.06Divide by 9.80: 4,217.06 /9.80≈430.1 meters.But wait, if the bullet is to be dropped from a height h, but the question asks for speed. So maybe they want the velocity equivalent? If we use kinetic energy instead of potential energy. Wait, if it's dropped from height h, then the velocity at impact is sqrt(2gh). But if they want the speed needed to achieve that temperature rise upon impact, regardless of height, then we can use the kinetic energy.Wait, the problem says "What speed would the bullet need to be dropped from above to raise its temperature to the melting point of lead". So maybe the speed just before impact, which would correspond to being dropped from a certain height. So using the same approach: the kinetic energy (1/2 mv²) is converted into heat Q= KE /E, which is then Q= m*C*ΔT. Therefore:(1/2 mv²)/E =m*C*ΔT ⇒v²/(2E)=C*ΔT ⇒v= sqrt(2E*C*ΔT)Plugging in the numbers:E=425, C=0.0315, ΔT=315°Cv= sqrt(2*425*0.0315*315)First compute 2*425=850850*0.0315≈26.77526.775*315≈26.775*300=8,032.5; 26.775*15=401.625; total≈8,434.125Then v= sqrt(8,434.125)≈91.84 m/sAlternatively, precise calculation:But let's compute step by step:2*425=850850*0.0315=850*0.03 +850*0.0015=25.5 +1.275=26.77526.775*315=26.775*(300+15)=26.775*300=8,032.5 +26.775*15=401.625=8,434.125sqrt(8,434.125). Let's see:90^2=8,10091^2=8,28192^2=8,464So sqrt(8,434.125) is between 91 and 92.Compute 91.8^2= (90 +1.8)^2=90^2 +2*90*1.8 +1.8^2=8,100 +324 +3.24=8,427.2491.8^2=8,427.2491.84^2= (91.8 +0.04)^2=91.8^2 +2*91.8*0.04 +0.04^2=8,427.24 +7.344 +0.0016≈8,434.5856But our target is 8,434.125. So 91.84^2≈8,434.59, which is slightly above. So the exact value is a bit less than 91.84. Let's do linear approximation.Between 91.8 and 91.84:At 91.8: 8,427.24At 91.84:8,434.5856The difference between 8,434.125 and8,427.24 is 6.885.From 91.8 to91.84, which is 0.04 seconds, the value increases by7.3456.So to get an increase of6.885, the fraction is6.885 /7.3456≈0.937Thus, the speed is91.8 +0.04*0.937≈91.8 +0.0375≈91.8375 m/s≈91.84 m/sSo approximately91.8 m/s. But let's check if this corresponds to a height.If v= sqrt(2gh), then h= v²/(2g)= (91.8)^2/(2*9.80)=8,427.24/(19.6)≈430 meters, which matches the height calculated earlier. So either the required speed is91.8 m/s, or the height is430 m. But the question asks for speed, so it's91.8 m/s. But let's check the calculation again.Wait, in the first question, we found that falling from100 m gives a temperature rise of73.2°C. So to get a temperature rise of315°C, which is315/73.2≈4.3 times higher, so height needed is4.3*100≈430 m. Which matches the calculation. So the speed is sqrt(2gh)=sqrt(2*9.8*430)=sqrt(8428)=91.8 m/s. So both methods give the same answer.But the problem states "What speed would the bullet need to be dropped from above to raise its temperature to the melting point of lead". So speed just before impact is91.8 m/s. But if they ask for the speed it needs to be "dropped from above", which is the same as the speed at impact. So answer is approximately91.8 m/s. But let's present it with more precision or as per the calculation.Original calculation:v= sqrt(2*425*0.0315*315)=sqrt(8,434.125)=91.84 m/s. So we can write≈91.8 m/s or92 m/s. Given the data, three significant figures would be91.8 m/s.Alternatively, if the problem expects the answer in terms of being dropped from a height, but they asked for speed, so it's the impact speed. So the answer is approximately91.8 m/s.Therefore, answers:1. 93.2°C2.91.8 m/sBut let me confirm if the units for specific heat and mechanical equivalent are correctly handled. Since in the first calculation, using E=425 and C=0.0315 with units that cancel appropriately gives the correct temperature rise. The key is that using the formula ΔT= gh/(E*C) gives the temperature rise without needing to know the mass, which cancels out.For the second part, using KE instead of PE. Since if the bullet is to be dropped, the KE at impact is equal to the PE at height h. So whether we use mgh or (1/2)mv², they are equal, so mgh=(1/2)mv² ⇒v= sqrt(2gh). Therefore, the calculation of the speed can be done either way, and since we found h=430 m, v= sqrt(2*9.8*430)=91.8 m/s. Alternatively, using the energy directly:Heat required Q= m*C*ΔT= m*0.0315*315=9.9225 mMechanical energy needed= Q*E=9.9225 m *425=4217.06 mSet equal to kinetic energy: (1/2)mv²=4217.06 m ⇒v²=8434.12 ⇒v≈91.84 m/s.Yes, same result.So both answers are:1. 93.2°C2.91.8 m/sBut need to check if the problem requires units. For temperature, degrees Celsius, which is specified. For speed, m/s, which is consistent with the given g=9.80 m/s².Alternatively, if in the second question, they ask for the height, but the question specifically says "What speed would the bullet need to be dropped from above", so the answer is the speed upon impact, which is91.8 m/s. So that's the answer.Final Answer1. The temperature of the bullet immediately after the fall is boxed{93.2^circ text{C}}.2. The speed needed to raise its temperature to the melting point of lead is boxed{91.8 dfrac{text{m}}{text{s}}}.</think>To solve the given problem, we need to determine the temperature of the bullet after a 100 m fall and the required speed to raise its temperature to the melting point of lead. 1. Temperature of the Bullet After the Fall1. Potential Energy Conversion: The bullet's potential energy at height ( h ) is converted into heat energy. The potential energy is ( mgh ).2. Heat Energy Calculation: The heat absorbed by the bullet is given by ( Q = frac{mgh}{E} ), where ( E ) is the mechanical equivalent of heat.3. Temperature Rise: The heat energy ( Q ) is used to raise the bullet's temperature, ( Q = mCDelta T ). Equating the two expressions for ( Q ) and solving for ( Delta T ): [ Delta T = frac{gh}{EC} ]4. Substitute Values: Using ( g = 9.80 , text{m/s}^2 ), ( h = 100 , text{m} ), ( E = 425 ), and ( C = 0.0315 ): [ Delta T = frac{9.80 times 100}{425 times 0.0315} approx 73.2^circ text{C} ]5. Final Temperature: The initial temperature is ( 20^circ text{C} ), so the final temperature is: [ 20 + 73.2 = 93.2^circ text{C} ] 2. Speed Required to Reach Melting Point1. Temperature Rise Needed: The required temperature rise is ( 335^circ text{C} - 20^circ text{C} = 315^circ text{C} ).2. Energy Conversion: The kinetic energy required to achieve this temperature rise is converted into heat. The kinetic energy is ( frac{1}{2}mv^2 ).3. Heat Energy Calculation: The heat energy ( Q = frac{frac{1}{2}mv^2}{E} ). Equating this to the heat required for the temperature rise: [ frac{1}{2}v^2 = ECDelta T ]4. Solve for Speed: [ v = sqrt{2ECDelta T} ]5. Substitute Values: Using ( E = 425 ), ( C = 0.0315 ), and ( Delta T = 315^circ text{C} ): [ v = sqrt{2 times 425 times 0.0315 times 315} approx 91.8 , text{m/s} ] Final Answers1. The temperature of the bullet immediately after the fall is boxed{93.2^circ text{C}}.2. The speed needed to raise its temperature to the melting point of lead is boxed{91.8 dfrac{text{m}}{text{s}}}.]

🔑:Let the initial temperature be t, the temperature after the fall be t_{1}, the mass of the lead m, and the path of the falling body s.It is necessary that the lost potential energy be equal to the heat gain:m g s=left(t_{1}-tright) m E Cfrom whicht_{1}=t+frac{g s}{E C}=93.2^{circ}If in the second case we want the heat gain to be greater, the body must be given an initial velocity v_{0}; in this case:m g s+frac{m v_{0}^{2}}{2}=(T-t) m E Cfrom whichv_{0}=sqrt{2[(T-t) E C-g s]}=80.46 frac{mathrm{m}}{mathrm{sec}}(Riesz Frigyes, Gymn. VIII. o. t., Györ.)The problem was also solved by: Szabó István.

🧠:[<think>Okay, so I need to figure out which of these statements about the correlation coefficient, r, is correct. Let's start by recalling what I know about the correlation coefficient. First, the correlation coefficient, often denoted by r, measures the strength and direction of a linear relationship between two variables. The value of r ranges from -1 to 1. So, the absolute value of r, |r|, must be between 0 and 1. That means |r| is always less than or equal to 1. So any options that say |r| ≥ 1 are probably incorrect. Let me check the options again.Option A says |r| ≤ 1, which is correct. Then it states that the closer |r| is to 0, the greater the degree of correlation, and closer to 1 means smaller. Wait, that doesn't sound right. Because I remember that when |r| is close to 1, it indicates a strong correlation, either positive or negative, while when |r| is close to 0, it means a weak or no correlation. So Option A is actually reversing the relationship. So A is incorrect.Option B claims |r| ≥ 1. That's wrong right off the bat because the correlation coefficient can't exceed 1 or be less than -1. So B is out.Option C says |r| ≤ 1, which is correct. Then it states the closer |r| is to 1, the greater the degree of correlation; closer to 0, the smaller. That aligns with what I remember. When the absolute value of r is near 1, it's a strong correlation, and near 0 is weak. So Option C seems correct so far.Option D again says |r| ≥ 1, which is false. So D is out.So the possible correct answer is C. Let me just verify once more. The correlation coefficient's magnitude (absolute value) indicates the strength of the relationship. So higher |r| (closer to 1) means stronger correlation, lower |r| (closer to 0) means weaker. So yes, C is correct. The other options either have the inequality wrong or the interpretation reversed.Final Answerboxed{C}</think>The correlation coefficient ( r ) measures the strength and direction of a linear relationship between two variables. The value of ( r ) ranges from -1 to 1, so ( |r| leq 1 ). - Option A states that the closer ( |r| ) is to 0, the greater the degree of correlation, which is incorrect because a value closer to 1 indicates a stronger correlation.- Option B and Option D incorrectly claim ( |r| geq 1 ), which is false since ( |r| ) cannot exceed 1.- Option C correctly states ( |r| leq 1 ) and that the closer ( |r| ) is to 1, the greater the degree of correlation, while the closer ( |r| ) is to 0, the smaller the degree of correlation.Thus, the correct answer is (boxed{C}).,<think>Okay, let's try to figure out which of these statements about the correlation coefficient r is correct. Hmm, I remember that the correlation coefficient measures how strong the linear relationship is between two variables. The value of r ranges from -1 to 1, right? So the absolute value of r, |r|, must be between 0 and 1. That means |r| can't be greater than 1. So looking at the options, statements B and D say |r| ≥ 1, which must be wrong because the correlation coefficient can't be more than 1 or less than -1. So we can eliminate options B and D right away.Now, between options A and C. Both A and C say |r| ≤ 1, which is correct. The difference is in how they describe the degree of correlation. Option A says the closer |r| is to 0, the greater the degree of correlation, and closer to 1, the smaller. But that doesn't sound right. Wait, if |r| is close to 1, it means a strong correlation, either positive or negative. If |r| is close to 0, it means a weak correlation. So actually, the closer |r| is to 1, the stronger the correlation, and closer to 0, the weaker.So looking at option C: "the closer |r| is to 1, the greater the degree of correlation; the closer |r| is to 0, the smaller the degree of correlation." That seems correct. Option A is the opposite, so A is wrong. Therefore, the correct answer should be C.Let me just double-check. The correlation coefficient's magnitude indicates the strength, with 1 being a perfect correlation and 0 being no correlation. So higher absolute values mean stronger relationships. Yep, so C is definitely correct. The others have either the wrong range for |r| or invert the relationship between |r| and correlation strength. Yeah, I think that's right.Final Answerboxed{C}</think>The correlation coefficient ( r ) measures the strength and direction of a linear relationship between two variables. The value of ( r ) ranges from -1 to 1, so ( |r| leq 1 ). This eliminates options B and D, which incorrectly state ( |r| geq 1 ).Next, we consider the remaining options A and C. The correlation strength is determined by how close ( |r| ) is to 1. A value of ( |r| ) close to 1 indicates a strong correlation, while a value close to 0 indicates a weak correlation. Therefore, option C correctly states that the closer ( |r| ) is to 1, the greater the degree of correlation, and the closer ( |r| ) is to 0, the smaller the degree of correlation. Option A incorrectly inverts this relationship.Thus, the correct answer is (boxed{C}).]

🔑:The correlation coefficient r can be used to measure the strength of the relationship between two variables. The closer the absolute value of r is to 1, the stronger the linear correlation between the two variables. When the absolute value of r is close to 0, it indicates that there is almost no correlation between the two variables. Therefore, the statement "For the correlation coefficient r, |r| leq 1, the closer |r| is to 1, the greater the degree of correlation; the closer |r| is to 0, the smaller the degree of correlation" is correct. Hence, the correct choice is: boxed{C}The correlation coefficient r can measure the strength of the relationship between two variables, with the absolute value of r closer to 1 indicating a stronger linear correlation. The range of the correlation coefficient is [-1, 1], leading to the conclusion. This question tests the ability to judge the strength of the linear correlation between two variables based on the correlation coefficient, and it is a basic question.