🧠:[<think>Okay, let's try to solve this problem step by step. So, we have a regular triangular prism ABC-A₁B₁C₁. The base edges are length 1, and the height of the prism is 2. The center P of the upper base triangle A₁B₁C₁ is given. Then, there's a plane through the lower base edge BC that is perpendicular to the line AP. This plane intersects the edge AA₁ at point D. We need to find the area of the section triangle BCD.First, let me visualize the prism. A regular triangular prism has two congruent equilateral triangles as bases connected by rectangles. The base edge length is 1, so each side of the triangle ABC is 1 unit. The height of the prism is 2, meaning the distance between the bases ABC and A₁B₁C₁ is 2 units.Point P is the center of the upper base triangle A₁B₁C₁. Since it's a regular triangle, the center should be the centroid, right? The centroid is the intersection point of the medians. In an equilateral triangle, the centroid is also the circumcenter and incenter, so it's located at a distance of one-third the height from each side.But wait, since the base is A₁B₁C₁, the coordinates of P would be the average of the coordinates of A₁, B₁, and C₁. Maybe it's easier if I assign coordinates to each point to make this more concrete.Let me set up a coordinate system. Let's place the lower base triangle ABC on the xy-plane with point A at the origin. Let me define the coordinates:- Let’s place point A at (0, 0, 0).- Since ABC is an equilateral triangle with side length 1, point B can be at (1, 0, 0).- To find point C, in the xy-plane, the coordinates would be (0.5, (√3)/2, 0) because the height of an equilateral triangle with side 1 is (√3)/2.Now, the upper base A₁B₁C₁ is directly above ABC with a height of 2. So:- A₁ is at (0, 0, 2)- B₁ is at (1, 0, 2)- C₁ is at (0.5, (√3)/2, 2)Then, the centroid P of triangle A₁B₁C₁ is the average of their coordinates:P_x = (0 + 1 + 0.5)/3 = 1.5/3 = 0.5P_y = (0 + 0 + (√3)/2)/3 = (√3)/6 ≈ 0.2887P_z = (2 + 2 + 2)/3 = 6/3 = 2Wait, hold on. That can't be right. Because each of A₁, B₁, C₁ is at z=2, so their centroid P must also be at z=2. So, the coordinates of P would be (0.5, (√3)/6, 2). Hmm, right, because the centroid's coordinates are the average of the three vertices.Wait, let's check that again. For triangle A₁B₁C₁:A₁ is (0,0,2), B₁ is (1,0,2), C₁ is (0.5, (√3)/2,2). So, centroid P has x-coordinate (0 + 1 + 0.5)/3 = 1.5/3 = 0.5, y-coordinate (0 + 0 + (√3)/2)/3 = (√3)/6, z-coordinate (2 + 2 + 2)/3 = 2. So yes, P is (0.5, √3/6, 2). That makes sense.Now, the line AP connects point A (0,0,0) to point P (0.5, √3/6, 2). Let's find the parametric equations for line AP.Parametric equations for a line through points A(0,0,0) and P(0.5, √3/6, 2):x(t) = 0 + t*(0.5 - 0) = 0.5ty(t) = 0 + t*(√3/6 - 0) = (√3/6)tz(t) = 0 + t*(2 - 0) = 2tSo, AP is parameterized as (0.5t, (√3/6)t, 2t), where t ranges from 0 to 1.Now, we need a plane that passes through edge BC (which is on the lower base ABC) and is perpendicular to line AP. This plane intersects edge AA₁ at point D. The section created is triangle BCD, and we need to find its area.First, let's confirm the coordinates of points B, C, and D.Points B and C are on the lower base:- B is (1, 0, 0)- C is (0.5, (√3)/2, 0)Edge BC is from (1,0,0) to (0.5, √3/2, 0). The plane in question passes through this edge BC and is perpendicular to line AP. Then, it intersects edge AA₁ at D.Edge AA₁ goes from A(0,0,0) to A₁(0,0,2). So, any point D on AA₁ has coordinates (0,0,k) where k is between 0 and 2. But since the plane passes through BC and D, which is on AA₁, we need to find the value of k where the plane passes through (0,0,k), BC, and is perpendicular to AP.To find the equation of the plane, since it is perpendicular to AP, the normal vector of the plane should be in the direction of AP. The direction vector of AP is from A to P, which is (0.5, √3/6, 2). Therefore, the normal vector of the plane is (0.5, √3/6, 2).But wait, planes perpendicular to a line have their normal vector parallel to that line. So, yes, the plane's normal vector is the direction vector of AP.But we also know that the plane passes through points B, C, and D. Since B and C are on the lower base (z=0), and D is on AA₁ (so D is (0,0,k) for some k). Therefore, the plane passes through points B(1,0,0), C(0.5, √3/2, 0), and D(0,0,k). Let's use these three points to find the equation of the plane and also ensure that it is perpendicular to AP.First, let's find the equation of the plane passing through points B(1,0,0), C(0.5, √3/2, 0), and D(0,0,k). Since two of these points (B and C) are on the lower base (z=0) and D is on the z-axis (x=0, y=0, z=k), the plane must pass through these three points.Alternatively, since the plane is perpendicular to AP, its normal vector is the direction vector of AP. So, the normal vector is (0.5, √3/6, 2). Let's use this to write the plane equation.The general equation of a plane with normal vector (a, b, c) is a(x - x₀) + b(y - y₀) + c(z - z₀) = 0, where (x₀, y₀, z₀) is a point on the plane.We can use point B(1,0,0) as the point on the plane. So, substituting:0.5(x - 1) + (√3/6)(y - 0) + 2(z - 0) = 0Simplify:0.5x - 0.5 + (√3/6)y + 2z = 0Rearranged:0.5x + (√3/6)y + 2z = 0.5Multiply both sides by 6 to eliminate denominators:3x + √3 y + 12 z = 3So, the equation of the plane is 3x + √3 y + 12 z = 3.Wait, but we also need this plane to pass through point D(0,0,k). Let's verify by plugging D into the equation:3*0 + √3*0 + 12*k = 3 ⇒ 12k = 3 ⇒ k = 3/12 = 1/4.Therefore, point D is at (0,0,1/4). Wait, but AA₁ is from (0,0,0) to (0,0,2), so z-coordinate is 1/4, which is 0.25. So, D divides AA₁ in the ratio 1/4 : 7/4.But let me check this again. The plane passes through B, C, and D, and is perpendicular to AP. So, according to the calculation above, the plane equation is 3x + √3 y + 12 z = 3, and point D is (0,0,1/4). So, point D is (0,0,0.25). Therefore, the coordinates of D are (0,0,1/4). Then, the section is triangle BCD with vertices at B(1,0,0), C(0.5, √3/2,0), and D(0,0,1/4).Now, we need to find the area of triangle BCD. Since it's a triangle in 3D space, one way to compute its area is to use vectors and compute the cross product.First, let's find vectors from point B to C and from point B to D.Vector BC = C - B = (0.5 - 1, (√3)/2 - 0, 0 - 0) = (-0.5, √3/2, 0)Vector BD = D - B = (0 - 1, 0 - 0, 1/4 - 0) = (-1, 0, 1/4)Then, the area is (1/2) the magnitude of the cross product of BC and BD.Compute the cross product BC × BD:BC = (-0.5, √3/2, 0)BD = (-1, 0, 1/4)Cross product is determinant:i | j | k-0.5 | √3/2 | 0-1 | 0 | 1/4Calculating determinant:i*( (√3/2)(1/4) - 0*0 ) - j*( (-0.5)(1/4) - 0*(-1) ) + k*( (-0.5)(0) - (√3/2)(-1) )Compute each component:i: (√3/2 * 1/4) = √3/8j: - [ (-0.5 * 1/4) - 0 ] = - ( -1/8 ) = 1/8k: (0 - (-√3/2)) = √3/2So, the cross product is (√3/8, 1/8, √3/2)Now, the magnitude of this cross product is sqrt( (√3/8)^2 + (1/8)^2 + (√3/2)^2 )Compute each term:(√3/8)^2 = (3)/64(1/8)^2 = 1/64(√3/2)^2 = 3/4Sum: 3/64 + 1/64 + 3/4 = (4/64) + (48/64) = 52/64 = 13/16Therefore, magnitude is sqrt(13/16) = (√13)/4Therefore, the area of triangle BCD is (1/2)*(√13)/4 = √13/8Wait, but let me check the cross product again. Because when I computed the cross product components:i component: (√3/2)(1/4) - 0*0 = √3/8j component: - [ (-0.5)(1/4) - 0*(-1) ] = - [ (-0.5/4) - 0 ] = - [ -1/8 ] = 1/8k component: (-0.5)(0) - (√3/2)(-1) = 0 + √3/2 = √3/2So cross product vector is (√3/8, 1/8, √3/2). Then, the magnitude squared is ( (√3/8)^2 + (1/8)^2 + (√3/2)^2 ) = 3/64 + 1/64 + 3/4 = (4/64) + (48/64) = 52/64 = 13/16. So sqrt(13/16) = √13 /4. Then, half of that is √13 /8. So the area is √13 divided by 8. Hmm, but let's see if that makes sense.Alternatively, maybe there's a mistake in the cross product. Let me double-check.Alternatively, perhaps using coordinates for the triangle points and computing the area with the formula for a triangle in 3D space.Coordinates of B: (1, 0, 0)Coordinates of C: (0.5, √3/2, 0)Coordinates of D: (0, 0, 1/4)We can use the formula for the area of a triangle given by three points in space:Area = (1/2) | (BC × BD) |, which we did, so that should be correct. So if the cross product is calculated correctly, then the area is √13 /8.But let's see, the answer seems a bit small, but considering the coordinates, maybe it's correct. Let me check another way.Alternatively, perhaps we can find the lengths of the sides of triangle BCD and use Heron's formula.First, compute the distances between the three points.Distance BC: since B and C are on the base, distance BC is 1 (since it's a base edge). But wait, in the coordinates:B is (1,0,0), C is (0.5, √3/2,0). The distance between B and C is sqrt( (0.5 - 1)^2 + (√3/2 - 0)^2 + 0 ) = sqrt( (-0.5)^2 + (√3/2)^2 ) = sqrt(0.25 + 0.75) = sqrt(1) = 1. So BC is 1.Distance BD: from B(1,0,0) to D(0,0,1/4). Distance is sqrt( (0 - 1)^2 + (0 - 0)^2 + (1/4 - 0)^2 ) = sqrt(1 + 0 + 1/16 ) = sqrt(17/16) = √17 /4 ≈ 1.030776406Distance CD: from C(0.5, √3/2,0) to D(0,0,1/4). Distance is sqrt( (0 - 0.5)^2 + (0 - √3/2)^2 + (1/4 - 0)^2 ) = sqrt(0.25 + 3/4 + 1/16) = sqrt(1 + 1/16) = sqrt(17/16) = √17 /4 ≈ 1.030776406So triangle BCD has sides BC =1, BD=CD=√17 /4. So it's an isoceles triangle with two sides equal to √17 /4 and base 1.Using Heron's formula, semi-perimeter s = (1 + √17/4 + √17/4 ) /2 = (1 + √17/2)/2 = (2 + √17)/4Area = sqrt( s(s - a)(s - b)(s - c) )s - a = (2 + √17)/4 - 1 = (2 + √17 -4)/4 = (-2 + √17)/4s - b = (2 + √17)/4 - √17/4 = (2 + √17 - √17)/4 = 2/4 = 1/2s - c = same as s - b, since b and c are equal. So 1/2Therefore, Area = sqrt( [ (2 + √17)/4 ] * [ (-2 + √17)/4 ] * [1/2] * [1/2] )First compute [ (2 + √17)/4 ] * [ (-2 + √17)/4 ] = [ (√17)^2 - (2)^2 ] / (16) = (17 -4)/16 =13/16Then, [1/2] * [1/2] =1/4So overall:sqrt( (13/16)*(1/4) ) = sqrt(13/64 ) = √13 /8Which matches the previous result. Therefore, the area is indeed √13 /8. So the answer should be boxed{dfrac{sqrt{13}}{8}}.But let me verify once again if point D is correctly found. Because in the plane equation, we found that D is (0,0,1/4). Let me confirm that.We had the plane equation 3x + √3 y + 12 z = 3. Plugging in D(0,0,k):3*0 + √3*0 +12*k =3 => 12k=3 =>k=1/4. Yes, that's correct. So D is indeed at (0,0,1/4). So the coordinates are correct.Alternatively, maybe using vectors to find where AP is perpendicular to the plane. Since the plane is perpendicular to AP, then the direction vector of AP is the normal to the plane. So, the plane equation can be written as 0.5x + (√3/6)y + 2z = d. To find d, since the plane passes through point B(1,0,0):0.5*1 + (√3/6)*0 +2*0 = d => d=0.5. Therefore, the plane equation is 0.5x + (√3/6)y + 2z = 0.5. Multiply both sides by 6 to eliminate fractions:3x + √3 y +12 z = 3. Which matches our previous equation. Then, point D is (0,0,1/4) as above. So all steps check out.Therefore, the area of triangle BCD is √13 /8, which is approximately 0.489. Given the prism's dimensions, this seems reasonable.Alternatively, maybe another approach using projections. Since the plane is perpendicular to AP, perhaps the triangle BCD is a right triangle or something, but given the cross product result, it's not a right triangle. The Heron's formula approach confirmed the area.Therefore, after multiple checks, I think the area is indeed √13 divided by 8.Final Answerboxed{dfrac{sqrt{13}}{8}}</think>Given a regular triangular prism (ABC-A_1B_1C_1) with a height of 2 and a base edge length of 1, we need to find the area of the section (triangle BCD) formed by a plane through the lower base edge (BC) that is perpendicular to the line (AP), where (P) is the center of the upper base (triangle A_1B_1C_1).1. Coordinates Setup: - Lower base vertices: (A(0, 0, 0)), (B(1, 0, 0)), (C(0.5, sqrt{3}/2, 0)). - Upper base vertices: (A_1(0, 0, 2)), (B_1(1, 0, 2)), (C_1(0.5, sqrt{3}/2, 2)). - Center (P) of the upper base: (P(0.5, sqrt{3}/6, 2)).2. Line (AP) Parametrization: - Parametric equations: (x(t) = 0.5t), (y(t) = (sqrt{3}/6)t), (z(t) = 2t).3. Plane Equation: - The plane is perpendicular to (AP), so its normal vector is the direction vector of (AP): ((0.5, sqrt{3}/6, 2)). - Plane equation passing through (B(1, 0, 0)): (0.5(x - 1) + (sqrt{3}/6)y + 2z = 0). - Simplifies to: (3x + sqrt{3}y + 12z = 3).4. Finding Point (D): - Point (D) on edge (AA_1) has coordinates ((0, 0, k)). - Substituting into the plane equation: (12k = 3 Rightarrow k = 1/4). Thus, (D(0, 0, 1/4)).5. Area Calculation: - Vectors from (B) to (C) and (B) to (D): (overrightarrow{BC} = (-0.5, sqrt{3}/2, 0)), (overrightarrow{BD} = (-1, 0, 1/4)). - Cross product (overrightarrow{BC} times overrightarrow{BD} = (sqrt{3}/8, 1/8, sqrt{3}/2)). - Magnitude of the cross product: (sqrt{(sqrt{3}/8)^2 + (1/8)^2 + (sqrt{3}/2)^2} = sqrt{13}/4). - Area of (triangle BCD): (frac{1}{2} times sqrt{13}/4 = sqrt{13}/8).The area of the section (triangle BCD) is (boxed{dfrac{sqrt{13}}{8}}).,<think>Okay, so I need to find the area of the triangle BCD in this regular triangular prism. Let me first try to visualize the prism. It's a regular triangular prism, which means both bases are equilateral triangles with edge length 1, and the height of the prism (the distance between the two bases) is 2. The centers of the upper and lower bases are points P and maybe another point, but the problem mentions P as the center of the upper base, which is triangle A1B1C1. The plane BCD is perpendicular to the line AP, and it goes through the lower base edge BC. This plane intersects the edge AA1 at point D. So, the section we're supposed to find the area of is triangle BCD. First, let me get the coordinates of all the points to make this easier. Maybe setting up a coordinate system will help. Let me place the lower base triangle ABC in the xy-plane. Let me assign coordinates:Let’s put point A at (0, 0, 0), point B at (1, 0, 0), and point C at (0.5, √3/2, 0). Since the base is an equilateral triangle with edge length 1, the coordinates should be correct. Then the upper base triangle A1B1C1 will be at z = 2. So, A1 is (0, 0, 2), B1 is (1, 0, 2), and C1 is (0.5, √3/2, 2). The center P of the upper base is the centroid of triangle A1B1C1. The centroid of a triangle is the average of its vertices' coordinates. So, the coordinates of P would be ((0 + 1 + 0.5)/3, (0 + 0 + √3/2)/3, (2 + 2 + 2)/3) = (1.5/3, (√3/2)/3, 6/3) = (0.5, √3/6, 2). Wait, but actually, since all the upper base points have z-coordinate 2, the centroid should have z-coordinate 2. But the centroid coordinates for the x and y would be the average of the x and y coordinates of A1, B1, C1. A1 is (0,0,2), B1 is (1,0,2), and C1 is (0.5, √3/2, 2). So, x-coordinates: 0, 1, 0.5. Average is (0 + 1 + 0.5)/3 = 1.5/3 = 0.5. y-coordinates: 0, 0, √3/2. Average is (0 + 0 + √3/2)/3 = (√3/2)/3 = √3/6. So, centroid P is (0.5, √3/6, 2). Okay, now point A is (0,0,0). So the line AP connects point A (0,0,0) to point P (0.5, √3/6, 2). Let me find the parametric equations for line AP. The direction vector from A to P is (0.5 - 0, √3/6 - 0, 2 - 0) = (0.5, √3/6, 2). So parametric equations are:x = 0 + 0.5*ty = 0 + (√3/6)*tz = 0 + 2*twhere t ranges from 0 to 1.So line AP is parametrized as (0.5t, (√3/6)t, 2t).Now, the plane BCD is supposed to be perpendicular to line AP and passes through edge BC. Edge BC is from point B (1,0,0) to point C (0.5, √3/2, 0). So edge BC is in the lower base, lying in the xy-plane (z=0). Since the plane passes through BC and is perpendicular to AP, we need to find the equation of this plane. To find the equation of the plane, we need a point on the plane and a normal vector. Since the plane is perpendicular to AP, the direction vector of AP is the normal vector of the plane. The direction vector of AP is (0.5, √3/6, 2). So the normal vector n of the plane is (0.5, √3/6, 2). But maybe it's better to use a scalar multiple to make calculations easier. Let's multiply all components by 6 to eliminate denominators. Then the normal vector becomes (3, √3, 12). But perhaps we can just use the original components.Given that the plane passes through edge BC, which is from B (1,0,0) to C (0.5, √3/2, 0). Let's take point B (1,0,0) as a point on the plane. So the equation of the plane is:n ⋅ (r - r0) = 0where n is the normal vector, r is a general point on the plane, and r0 is point B.So the equation is:0.5(x - 1) + (√3/6)(y - 0) + 2(z - 0) = 0Simplify:0.5x - 0.5 + (√3/6)y + 2z = 0Multiply both sides by 6 to eliminate denominators:3x - 3 + √3 y + 12z = 0Then,3x + √3 y + 12z = 3So that's the equation of the plane BCD.Now, we need to find where this plane intersects the edge AA1. Edge AA1 goes from A (0,0,0) to A1 (0,0,2). Parametrize edge AA1 as:x = 0, y = 0, z = t, where t ∈ [0,2]So, substituting into the plane equation:3*0 + √3*0 + 12*t = 312t = 3t = 3/12 = 1/4So point D is (0, 0, 1/4). Wait, but edge AA1 is from (0,0,0) to (0,0,2), so at t = 1/4, the point is (0,0,1/4). Therefore, D is (0,0,1/4). But wait, that seems a bit low. Let me check the calculations again.Plane equation: 3x + √3 y + 12z = 3For edge AA1, x = 0, y = 0, z = t. So substituting:3*0 + √3*0 + 12t = 3 => 12t = 3 => t = 1/4. Yes, that's correct. So D is at (0,0,1/4). Now, we need to find the coordinates of points B, C, D to compute the area of triangle BCD.Point B is (1, 0, 0)Point C is (0.5, √3/2, 0)Point D is (0, 0, 1/4)So triangle BCD has vertices at these three points. To find its area, we can use the formula for the area of a triangle given by three points in 3D space. One way is to calculate the vectors from one point to the other two, compute the cross product of those vectors, and then take half the magnitude of the cross product.Let's take point B as the common vertex. Vectors BD and BC.Wait, actually, vectors from B to C and from B to D.Vector BC is C - B = (0.5 - 1, √3/2 - 0, 0 - 0) = (-0.5, √3/2, 0)Vector BD is D - B = (0 - 1, 0 - 0, 1/4 - 0) = (-1, 0, 1/4)Then the cross product of BC and BD is:|i     j         k     || -0.5 √3/2   0 || -1   0     1/4 |Calculating determinant:i * (√3/2 * 1/4 - 0 * 0) - j * (-0.5 * 1/4 - 0 * (-1)) + k * (-0.5 * 0 - √3/2 * (-1))= i * (√3/8) - j * (-0.5/4) + k * (√3/2)= (√3/8)i + (1/8)j + (√3/2)kThen the magnitude of this cross product is sqrt[(√3/8)^2 + (1/8)^2 + (√3/2)^2]Compute each component:(√3/8)^2 = (3)/64(1/8)^2 = 1/64(√3/2)^2 = 3/4Sum: 3/64 + 1/64 + 3/4 = (4/64) + (48/64) = 52/64 = 13/16Therefore, the magnitude is sqrt(13/16) = √13 /4Then the area is half of this: (√13)/8Wait, but let me verify that. Because the cross product magnitude is the area of the parallelogram, so half is the triangle area.But the cross product here is between BC and BD, giving the parallelogram area, so half of √13 /4 is √13 /8.But let's check the calculation again because this seems a bit small.Wait, let's compute each term step by step:First, cross product components:i component: √3/8j component: 1/8 (since it was -j*( -0.5/4 - 0 )) = -j*(-1/8) = j*(1/8)k component: √3/2 (from -0.5*0 - (-√3/2*1) = 0 + √3/2)So cross product vector is (√3/8, 1/8, √3/2)Compute the magnitude:sqrt[( (√3/8)^2 + (1/8)^2 + (√3/2)^2 )]Compute each squared term:(√3/8)^2 = 3/64(1/8)^2 = 1/64(√3/2)^2 = 3/4 = 48/64So sum: 3/64 + 1/64 + 48/64 = 52/64 = 13/16Therefore, magnitude is sqrt(13/16) = √13 /4Thus, area is half that: √13 /8 ≈ 0.49...But let me see if this makes sense. The base BC has length 1, since it's an edge of the base triangle. The height from D to BC would determine the area. But since D is at (0,0,1/4), and BC is in the z=0 plane, perhaps the height is the z-coordinate component? Wait, not exactly, because the triangle BCD is in 3D space, so the height isn't just the z-coordinate. Alternatively, maybe we can project the triangle onto a 2D plane where it's easier to compute. But the cross product method should be accurate. Let me confirm once more.Wait, let's compute the vectors again.Vector BC: from B(1,0,0) to C(0.5, √3/2, 0): (-0.5, √3/2, 0)Vector BD: from B(1,0,0) to D(0,0,1/4): (-1, 0, 1/4)Cross product components:i: (√3/2 * 1/4 - 0 * 0) = √3/8j: - [ (-0.5 * 1/4 - 0 * (-1)) ] = - [ (-1/8 - 0) ] = - (-1/8) = 1/8k: (-0.5 * 0 - √3/2 * (-1)) = 0 + √3/2 = √3/2Yes, that's correct. So cross product is (√3/8, 1/8, √3/2), magnitude squared is 3/64 + 1/64 + 3/4 = (4 + 48)/64 = 52/64, square root is √52 /8 = (2√13)/8 = √13 /4. Then half of that is √13 /8.Hmm, but maybe I made a mistake in the cross product calculation. Let me check another way. Let's compute the cross product using another method.Given vectors BC = (-0.5, √3/2, 0) and BD = (-1, 0, 1/4)Cross product BC × BD is:|i     j     k   ||-0.5 √3/2   0||-1   0   1/4|So calculate:i*( (√3/2)(1/4) - 0*0 ) - j*( (-0.5)(1/4) - 0*(-1) ) + k*( (-0.5)(0) - (√3/2)(-1) )Compute each term:i: (√3/2)(1/4) = √3/8j: - [ (-0.5)(1/4) - 0 ] = - [ -1/8 - 0 ] = - (-1/8) = 1/8k: (0 - (-√3/2)) = √3/2So same as before. So cross product is (√3/8, 1/8, √3/2). Magnitude squared:( (√3/8)^2 + (1/8)^2 + (√3/2)^2 ) = 3/64 + 1/64 + 3/4 = 3/64 + 1/64 + 48/64 = 52/64 = 13/16So magnitude is √13 /4, so area is √13 /8 ≈ 0.49.But let me think if this is reasonable. The base BC is length 1. The height from D to BC in 3D space would be the minimal distance from D to BC. Let's calculate that.The formula for the distance from a point to a line in 3D is |(D - B) × direction| / |direction|Here, direction vector of BC is (-0.5, √3/2, 0). The vector from B to D is BD = (-1, 0, 1/4). So cross product BD × direction (BC) is the same as BC × BD, which we already calculated as (√3/8, 1/8, √3/2). The magnitude is √13 /4.Wait, the distance from D to BC is |BD × BC| / |BC|So |BD × BC| is √13 /4, and |BC| is the length of BC, which is 1 (since BC is an edge of the base triangle). Wait, BC is length 1? Let's check:Point B is (1,0,0), point C is (0.5, √3/2, 0). So distance BC is sqrt( (0.5 -1)^2 + (√3/2 -0)^2 + 0 ) = sqrt( (-0.5)^2 + (√3/2)^2 ) = sqrt(0.25 + 0.75 ) = sqrt(1) = 1. Yes, so |BC| =1.Therefore, distance from D to BC is (√13 /4)/1 = √13 /4 ≈ 0.66.Then area is (base * height)/2 = (1 * √13 /4)/2 = √13 /8, which matches the previous result. So that's correct.Therefore, the area of triangle BCD is √13 /8.Wait, but the problem says "the area of the section triangle BCD". Let me check if the answer is expected to be in some simplified radical form or maybe a different expression. Alternatively, maybe I made a mistake in coordinates.Wait, let me check the coordinates again. Let me confirm the position of point D. The plane BCD intersects AA1 at D. When I found D at (0,0,1/4), that is, one fourth up from A towards A1. But since the prism's height is 2, so AA1 is of length 2, so D is at 1/4 of the way from A. That seems okay.Wait, but perhaps I should check the plane equation again. The plane passes through BC and is perpendicular to AP. So, the normal vector of the plane is the direction vector of AP. So direction vector AP is (0.5, √3/6, 2). Then, the plane equation is determined by this normal vector and passing through point B (1,0,0).Equation: 0.5(x -1) + (√3/6)(y - 0) + 2(z - 0) = 0Multiply through by 6 to eliminate denominators:3(x -1) + √3 y + 12 z = 0Which becomes:3x -3 + √3 y +12 z = 0Or 3x + √3 y +12 z = 3. Yes, that's correct.Then substituting edge AA1 (x=0, y=0, z=t) gives 12 t =3, so t=1/4, so D=(0,0,1/4). Correct.Therefore, coordinates are correct.Alternatively, maybe there is another way to compute the area. Let's compute the lengths of the sides of triangle BCD.Points:B (1,0,0)C (0.5, √3/2, 0)D (0,0,1/4)Compute the distances between each pair.BC: already known as 1.BD: distance from B to D: sqrt( (0 -1)^2 + (0 -0)^2 + (1/4 -0)^2 ) = sqrt(1 + 0 + 1/16 ) = sqrt(17/16) = √17 /4 ≈1.030CD: distance from C to D: sqrt( (0 -0.5)^2 + (0 - √3/2)^2 + (1/4 -0)^2 )= sqrt( 0.25 + 3/4 + 1/16 )= sqrt( 0.25 + 0.75 + 0.0625 ) = sqrt(1.0625 ) = sqrt(17/16 ) = √17 /4 ≈1.030So triangle BCD has sides of length 1, √17/4, √17/4. So it's an isoceles triangle with base 1 and equal sides √17/4.To compute the area, maybe using Heron's formula.Compute semi-perimeter: s = (1 + √17/4 + √17/4)/2 = (1 + √17/2)/2 = (2 + √17)/4But Heron's formula might get messy. Alternatively, since it's isoceles, we can compute the height.Let’s take the base BC of length 1. Then the height h can be computed from the Pythagorean theorem using the equal sides.Each equal side is √17/4, so:h^2 + (0.5)^2 = (√17/4)^2h^2 + 0.25 = 17/16h^2 = 17/16 - 4/16 = 13/16Therefore, h = √13 /4Thus, area is (base * height)/2 = (1 * √13 /4)/2 = √13 /8Same result as before. So this confirms the area is √13 /8.But let me check if this answer makes sense. Given that the prism height is 2 and the base edge is 1, the triangle BCD is a slanted triangle, and the area being √13 /8 ≈0.49. Since the base BC is 1, and the height is ≈0.66, which gives an area of ≈0.33, but wait, wait, no, using base 1 and height ≈0.66 would give area ≈0.33, but according to our calculation, it's ≈0.49. Wait, maybe my intuition is wrong here.Wait, no. The height here is not in the plane of the base. Since the triangle is in 3D space, the actual height isn't simply the vertical distance. The height is the minimal distance from D to BC, which we calculated as √13 /4 ≈0.66, so the area is (1 * 0.66)/2 ≈0.33. Wait, this contradicts the previous result. Wait, no, the problem here is that Heron's formula and cross product give the same result, but when I computed h from Heron's formula, I had h = √13 /4 ≈0.66, so area is 1*0.66 /2 ≈0.33. But according to cross product method, it's √13 /8 ≈0.49. Wait, this is a contradiction. Which one is correct?Wait, wait, no. Wait, Heron's formula: if the sides are 1, √17/4, √17/4.Compute semi-perimeter: (1 + √17/4 + √17/4)/2 = (1 + √17/2)/2 = 1/2 + √17/4 ≈0.5 +1.030≈1.530Then area = sqrt( s(s - a)(s - b)(s - c) )So s - a = 1.530 -1 ≈0.530s - b = s - √17/4 ≈1.530 -1.030≈0.5Similarly s - c = same as s - b ≈0.5Thus area ≈sqrt(1.530 *0.530 *0.5 *0.5 )Compute product inside sqrt:1.530 *0.530 ≈0.81090.5 *0.5 =0.25Total product: 0.8109 *0.25≈0.2027sqrt(0.2027 )≈0.450Which is roughly √13 /8≈0.49. Wait, but Heron's formula gives approximately 0.45, but according to cross product, it's approximately 0.49. There's a discrepancy. Which one is correct?Wait, actually, no. Wait, if the sides are a=1, b=√17/4≈1.030, c=√17/4≈1.030.Then semi-perimeter s=(1 +1.030 +1.030)/2≈(3.06)/2≈1.53s -a=1.53 -1=0.53s -b=1.53 -1.03≈0.5s -c=0.5So area≈sqrt(1.53 *0.53 *0.5 *0.5 )Compute:1.53 *0.53≈0.81090.5 *0.5=0.25Total≈0.8109*0.25≈0.2027sqrt(0.2027 )≈0.450But cross product gives area≈√13 /8≈0.49Hmm, discrepancy here. So which is correct?Wait, maybe the error is in Heron's formula calculation. Let me compute Heron's formula more precisely.Compute s=(1 + √17/4 + √17/4)/2=(1 + √17/2)/2=1/2 + √17/4s=0.5 + (√17)/4≈0.5 +1.030776406≈1.530776406s - a=1.530776406 -1≈0.530776406s - b=1.530776406 -√17/4≈1.530776406 -1.030776406≈0.5s - c=0.5Compute the product s(s -a)(s -b)(s -c)=1.530776406 *0.530776406 *0.5 *0.5First compute 1.530776406 *0.530776406≈(1.530776406 *0.5)+(1.530776406 *0.030776406)≈0.765388203 +0.047116≈0.812504Then multiply by 0.5 *0.5=0.25Total≈0.812504 *0.25≈0.203126Thus, sqrt(0.203126 )≈0.4506But cross product gives area as √13 /8≈0.4906So there's a discrepancy. Which one is correct?Wait, perhaps my mistake is in the assumption that the triangle has sides 1, √17/4, √17/4. Let me compute the lengths again.BD: distance from B(1,0,0) to D(0,0,1/4):sqrt( (0 -1)^2 + (0 -0)^2 + (1/4 -0)^2 )=sqrt(1 +0 +1/16)=sqrt(17/16)=√17/4≈1.030CD: distance from C(0.5, √3/2,0) to D(0,0,1/4):sqrt( (0 -0.5)^2 + (0 -√3/2)^2 + (1/4 -0)^2 )=sqrt(0.25 + 3/4 +1/16 )Convert to sixteenths:0.25=4/16, 3/4=12/16, 1/16=1/16Total: 4+12+1=17/16Thus sqrt(17/16)=√17/4≈1.030So BD and CD are both √17/4, BC is 1. So Heron's formula gives approximately 0.45, but cross product gives approximately 0.49. Contradiction.Therefore, there must be a mistake in one of the methods.Wait, but cross product is generally reliable in 3D. Let me recalculate the cross product area.Vectors BC and BD:BC: (-0.5, √3/2, 0)BD: (-1, 0, 1/4)Cross product:i*( (√3/2)(1/4) - 0*0 ) - j*( (-0.5)(1/4) - 0*(-1) ) + k*( (-0.5*0) - (√3/2)(-1) )= i*(√3/8) - j*(-1/8) + k*(√3/2)So cross product vector is (√3/8, 1/8, √3/2)Now, compute the magnitude:sqrt( (√3/8)^2 + (1/8)^2 + (√3/2)^2 )= sqrt( 3/64 + 1/64 + 3/4 )Convert 3/4 to 48/64:3/64 +1/64 +48/64=52/64=13/16So sqrt(13/16)=√13 /4≈3.6055/4≈0.901Then area is half of that:≈0.901/2≈0.450Wait, that contradicts my previous statement. Wait, wait, no:Wait, sqrt(13)/4≈3.6055/4≈0.901, but area is half that, so≈0.450, which matches Heron's formula. So where did I get √13 /8≈0.49 before? That was a mistake.Wait, sqrt(13)≈3.6055, so sqrt(13)/8≈0.4507, which is the same as Heron's formula. So yes, the cross product gives area≈0.4507, which is approximately equal to Heron's formula's 0.4506. So slight differences due to decimal approximations. So the correct area is sqrt(13)/8.Wait, but how?Wait, cross product magnitude is sqrt(13)/4, so area is half of that, so sqrt(13)/8≈0.4507. But earlier, when I computed the Heron's formula, I thought Heron's formula gives 0.45, and cross product gives 0.49, but that was a mistake. sqrt(13)/8≈0.4507, which is consistent with Heron's formula. So everything is correct.So the final answer is sqrt(13)/8. So boxed, it's boxed{dfrac{sqrt{13}}{8}}.But wait, let me check again.Alternatively, perhaps there was a miscalculation in cross product. Wait, the cross product vector is (√3/8, 1/8, √3/2). The magnitude is sqrt( (√3/8)^2 + (1/8)^2 + (√3/2)^2 )Compute each term:(√3/8)^2 = 3/64 ≈0.046875(1/8)^2 =1/64≈0.015625(√3/2)^2=3/4=0.75Total≈0.046875 +0.015625 +0.75=0.8125sqrt(0.8125 )≈0.901387819Then area is 0.901387819 /2≈0.450693909Which is exactly sqrt(13)/8≈0.450693909. Because sqrt(13)=3.605551275, so sqrt(13)/8≈0.450693909. Therefore, correct.Therefore, the area of triangle BCD is sqrt(13)/8.Thus, the answer is boxed{dfrac{sqrt{13}}{8}}.Final AnswerThe area of the section triangle BCD is boxed{dfrac{sqrt{13}}{8}}.</think>Given a regular triangular prism (ABC-A_1B_1C_1) with a height of 2 and a base edge length of 1, we need to find the area of the section (triangle BCD) where the plane (BCD) is perpendicular to (AP) and intersects the edge (AA_1) at point (D).1. Coordinates Setup: - Lower base vertices: (A(0, 0, 0)), (B(1, 0, 0)), (C(0.5, sqrt{3}/2, 0)). - Upper base vertices: (A_1(0, 0, 2)), (B_1(1, 0, 2)), (C_1(0.5, sqrt{3}/2, 2)). - Center (P) of the upper base: (P(0.5, sqrt{3}/6, 2)).2. Line (AP): - Parametric equations: (x = 0.5t), (y = (sqrt{3}/6)t), (z = 2t).3. Plane Equation: - Normal vector (direction of (AP)): ((0.5, sqrt{3}/6, 2)). - Plane through (B(1, 0, 0)): (0.5(x - 1) + (sqrt{3}/6)y + 2z = 0). - Simplified: (3x + sqrt{3}y + 12z = 3).4. Intersection with (AA_1): - Parametrize (AA_1): (x = 0), (y = 0), (z = t). - Substitute into plane equation: (12t = 3 Rightarrow t = 1/4). - Point (D(0, 0, 1/4)).5. Area Calculation: - Vectors (BC = (-0.5, sqrt{3}/2, 0)) and (BD = (-1, 0, 1/4)). - Cross product: ((sqrt{3}/8, 1/8, sqrt{3}/2)). - Magnitude: (sqrt{(sqrt{3}/8)^2 + (1/8)^2 + (sqrt{3}/2)^2} = sqrt{13}/4). - Area: (frac{1}{2} times sqrt{13}/4 = sqrt{13}/8).Thus, the area of the section (triangle BCD) is (boxed{dfrac{sqrt{13}}{8}}).]

🔑:2. frac{sqrt{13}}{8}.As shown in Figure 1, let the plane A A_{1} P intersect the edges B C and B_{1} C_{1} at points E and E_{1}, respectively.As shown in Figure 2, in the rectangle A E E_{1} A_{1}, it is easy to know that A A_{1}=2, A E=A_{1} E_{1}=frac{sqrt{3}}{2}, A_{1} P=frac{sqrt{3}}{3}. Let angle A_{1} A P=angle A E D=theta.Then tan theta=frac{A_{1} P}{A A_{1}}=frac{sqrt{3}}{6}.Thus, cos theta=sqrt{frac{12}{13}} Rightarrow D E=frac{A E}{cos theta}=frac{sqrt{13}}{4}Rightarrow S_{triangle B C D}=frac{1}{2} D E cdot B C=frac{sqrt{13}}{8}.

🧠:[<think>Okay, let's see. The problem says that if p and q are two different prime numbers, what's the remainder when p^{q-1} + q^{p-1} is divided by p times q. Hmm. So we need to find (p^{q-1} + q^{p-1}) mod (pq). First, I remember something about Fermat's Little Theorem. That theorem states that if you have a prime number, say p, then for any integer a not divisible by p, a^{p-1} ≡ 1 mod p. Similarly, a^{q-1} ≡ 1 mod q if a is not divisible by q. Wait, since p and q are primes, and they are different, so they are coprime. That might come in handy. Maybe we can use the Chinese Remainder Theorem here. To find the remainder modulo pq, it's enough to find the remainders modulo p and modulo q separately and then combine them. Let me try that approach. Let's first compute p^{q-1} + q^{p-1} modulo p. Since we're dealing with modulo p, any multiple of p will be congruent to 0. So let's see:First term: p^{q-1} mod p. Well, p is congruent to 0 mod p, so p^{q-1} is 0 mod p. Second term: q^{p-1} mod p. Since q is a prime different from p, it's not divisible by p. So by Fermat's Little Theorem, q^{p-1} ≡ 1 mod p. Therefore, the sum p^{q-1} + q^{p-1} ≡ 0 + 1 ≡ 1 mod p.Similarly, now let's compute the sum modulo q. First term: p^{q-1} mod q. Since p is a prime different from q, p is not divisible by q. So again, by Fermat's Little Theorem, p^{q-1} ≡ 1 mod q. Second term: q^{p-1} mod q. Since q is congruent to 0 mod q, this term is 0 mod q. Therefore, the sum p^{q-1} + q^{p-1} ≡ 1 + 0 ≡ 1 mod q.So now we have that the sum is congruent to 1 mod p and 1 mod q. Since p and q are coprime, by Chinese Remainder Theorem, there's a unique solution modulo pq. We need to find an integer x such that x ≡ 1 mod p and x ≡ 1 mod q. But 1 satisfies both congruences. Therefore, x ≡ 1 mod pq.Therefore, the remainder when p^{q-1} + q^{p-1} is divided by pq is 1. Wait, let me check this with an example to make sure. Let's take p = 2 and q = 3. Then p^{q-1} + q^{p-1} = 2^2 + 3^1 = 4 + 3 = 7. Divided by 2*3 = 6, the remainder is 1. Yep, that works. Another example: p = 3, q = 5. Then 3^{4} + 5^{2} = 81 + 25 = 106. Divided by 15, 15*7=105, so remainder 1. Again, 106 - 105 =1. So that works. Another example: p = 5, q = 7. Then 5^{6} + 7^{4} = 15625 + 2401 = 18026. Divided by 35. Let's see, 35*515 = 35*(500 + 15) = 17500 + 525 = 18025. So 18026 - 18025 =1. Remainder 1. Yep. Seems consistent. Therefore, the remainder is 1.Final AnswerThe remainder is boxed{1}.</think>Given two different prime numbers ( p ) and ( q ), we need to find the remainder when ( p^{q-1} + q^{p-1} ) is divided by ( p cdot q ).First, we use Fermat's Little Theorem, which states that for a prime ( p ) and an integer ( a ) not divisible by ( p ), ( a^{p-1} equiv 1 pmod{p} ). Similarly, for a prime ( q ) and an integer ( a ) not divisible by ( q ), ( a^{q-1} equiv 1 pmod{q} ).1. Modulo ( p ): - The term ( p^{q-1} equiv 0 pmod{p} ) because ( p ) is divisible by itself. - The term ( q^{p-1} equiv 1 pmod{p} ) by Fermat's Little Theorem since ( q ) is not divisible by ( p ). - Therefore, ( p^{q-1} + q^{p-1} equiv 0 + 1 equiv 1 pmod{p} ).2. Modulo ( q ): - The term ( p^{q-1} equiv 1 pmod{q} ) by Fermat's Little Theorem since ( p ) is not divisible by ( q ). - The term ( q^{p-1} equiv 0 pmod{q} ) because ( q ) is divisible by itself. - Therefore, ( p^{q-1} + q^{p-1} equiv 1 + 0 equiv 1 pmod{q} ).By the Chinese Remainder Theorem, since ( p ) and ( q ) are coprime, the system of congruences ( x equiv 1 pmod{p} ) and ( x equiv 1 pmod{q} ) has a unique solution modulo ( pq ). The solution is ( x equiv 1 pmod{pq} ).Thus, the remainder when ( p^{q-1} + q^{p-1} ) is divided by ( p cdot q ) is (boxed{1}).,<think>Okay, so I have this problem here: Let p and q be two different prime numbers. Find the remainder when p^{q-1} + q^{p-1} is divided by p*q. Hmm, interesting. Alright, let me try to figure this out step by step.First, primes... So p and q are primes, and they're different. So like, maybe 2 and 3, or 5 and 7, something like that. The expression is p^{q-1} + q^{p-1}, and we need to find the remainder when that's divided by p*q. So, essentially, compute (p^{q-1} + q^{p-1}) mod (p*q). Got it.I remember something about Fermat's Little Theorem. Let me recall. Fermat's Little Theorem states that if p is a prime and a is not divisible by p, then a^{p-1} ≡ 1 mod p. So maybe this theorem can help here since we have exponents of p and q.But here, we have p^{q-1} and q^{p-1}. Let me see. Let's take each term modulo p and modulo q separately. Then maybe use the Chinese Remainder Theorem? Wait, but since we need the result modulo p*q, perhaps combining the congruences?Wait, actually, if we can find what p^{q-1} + q^{p-1} is congruent to modulo p and modulo q, then we can combine those results to find the congruence modulo p*q. Because p and q are distinct primes, they're coprime, so Chinese Remainder Theorem applies. That sounds like a plan.So let's start with modulo p. Let's compute (p^{q-1} + q^{p-1}) mod p.First term: p^{q-1} mod p. Well, p is congruent to 0 mod p, so any power of p is also congruent to 0 mod p. So p^{q-1} ≡ 0 mod p.Second term: q^{p-1} mod p. Since q is a prime different from p, so q and p are coprime. Therefore, by Fermat's Little Theorem, q^{p-1} ≡ 1 mod p.So adding them together: 0 + 1 ≡ 1 mod p. So the entire expression is congruent to 1 mod p.Now let's do the same modulo q.First term: p^{q-1} mod q. Since p and q are distinct primes, p is coprime to q. So by Fermat's Little Theorem, p^{q-1} ≡ 1 mod q.Second term: q^{p-1} mod q. Similarly, q is congruent to 0 mod q, so any power of q is 0 mod q. Hence, q^{p-1} ≡ 0 mod q.Adding them together: 1 + 0 ≡ 1 mod q. So the expression is congruent to 1 mod q.Therefore, we have that p^{q-1} + q^{p-1} ≡ 1 mod p and ≡ 1 mod q. Since the expression is congruent to 1 modulo both p and q, and since p and q are coprime, by the Chinese Remainder Theorem, it must be congruent to 1 modulo p*q.Therefore, the remainder when p^{q-1} + q^{p-1} is divided by p*q is 1.Wait, let me check this with an example to make sure. Let's take p=2 and q=3, which are primes. Then compute p^{q-1} + q^{p-1} = 2^{2} + 3^{1} = 4 + 3 = 7. Then divide by p*q = 6. 7 divided by 6 leaves a remainder of 1. Yes, that works.Another example: p=3, q=5. Then 3^{4} + 5^{2} = 81 + 25 = 106. Divide by 15: 15*7=105, so 106-105=1. Remainder is 1. Hmm, that's the same result.Another one: p=5, q=7. 5^{6} + 7^{4} = 15625 + 2401 = 18026. Now, 5*7=35. Let's divide 18026 by 35. 35*515=35*(500 + 15)=17500 + 525=18025. So 18026 - 18025=1. Remainder is 1 again. Wow, so seems like 1 each time. So the reasoning seems correct.Therefore, the remainder is 1. So the answer should be 1. But let me just make sure that the Chinese Remainder Theorem step is correct.So if a number x ≡1 mod p and x ≡1 mod q, then x ≡1 mod (p*q). Because since x-1 is divisible by both p and q, and since p and q are coprime, x-1 is divisible by p*q. Hence, x ≡1 mod (p*q). Yes, that's the Chinese Remainder Theorem when the two congruences are the same. So that works.Therefore, the conclusion is correct. So the remainder is 1.Final AnswerThe remainder is boxed{1}.</think>Given two distinct prime numbers ( p ) and ( q ), we need to find the remainder when ( p^{q-1} + q^{p-1} ) is divided by ( p cdot q ).First, consider the expression modulo ( p ):- The term ( p^{q-1} equiv 0 mod p ) because any power of ( p ) is divisible by ( p ).- The term ( q^{p-1} equiv 1 mod p ) by Fermat's Little Theorem, since ( q ) is not divisible by ( p ).Thus, ( p^{q-1} + q^{p-1} equiv 0 + 1 equiv 1 mod p ).Next, consider the expression modulo ( q ):- The term ( p^{q-1} equiv 1 mod q ) by Fermat's Little Theorem, since ( p ) is not divisible by ( q ).- The term ( q^{p-1} equiv 0 mod q ) because any power of ( q ) is divisible by ( q ).Thus, ( p^{q-1} + q^{p-1} equiv 1 + 0 equiv 1 mod q ).Since ( p^{q-1} + q^{p-1} equiv 1 mod p ) and ( p^{q-1} + q^{p-1} equiv 1 mod q ), and ( p ) and ( q ) are coprime, by the Chinese Remainder Theorem, we conclude that ( p^{q-1} + q^{p-1} equiv 1 mod (p cdot q) ).Therefore, the remainder when ( p^{q-1} + q^{p-1} ) is divided by ( p cdot q ) is (boxed{1}).]

🔑:81 Prompt: Since p and q are different prime numbers, by Fermat's theorem, we know that p^{q-1} equiv 1(bmod q).Also, q^{p-1} equiv 0(bmod q), so p^{q-1}+q^{p-1} equiv 1(bmod q).Similarly, p^{q-1}+q^{p-1} equiv 1(bmod p).Since [p, q]=p q, it follows that p^{q-1}+q^{p-1} equiv 1(bmod p cdot q).

🧠:[<think>Okay, so I need to find the learning time required to reach a typing speed of 90 characters per minute using the given function t = -144 × log(1 - N/100). Let me break this down step by step.First, let's understand the function. The variable t represents the learning time in hours, and N is the typing speed in characters per minute. The problem is asking for t when N is 90. So essentially, I need to plug N = 90 into the equation and solve for t.Wait, let me make sure I have the function right. The user wrote it as t = -144 × lg (1 - N/100). Here, "lg" typically stands for logarithm base 10, right? So it's the common logarithm. But sometimes in some contexts, especially in computer science, "lg" can mean log base 2, but given the context here, I think it's more likely log base 10. Hmm, but I should check if that makes sense.Let me verify. If N is the typing speed, then 1 - N/100 would be 1 - 90/100 = 1 - 0.9 = 0.1 when N=90. So log(0.1) in base 10 is -1. Then multiplying by -144 gives t = -144 × (-1) = 144. So that would give t = 144 hours. That seems plausible, but let me make sure that I didn't misinterpret the logarithm base.Wait, but in some contexts, especially learning curves, they might use natural logarithm (ln) instead. But the problem uses "lg", which is more commonly log base 10. Let's check both possibilities just to be thorough.If "lg" is log base 10: log10(0.1) = -1, so t = -144 × (-1) = 144.If "lg" is natural logarithm (ln): ln(0.1) ≈ -2.302585, so t = -144 × (-2.302585) ≈ 144 × 2.302585 ≈ 331.57. That seems way too high for a learning time. So probably "lg" here is log base 10, as 144 hours for reaching 90 characters per minute is more reasonable, maybe?Alternatively, maybe it's log base 2? Let's see: log2(0.1) ≈ -3.321928, so t = -144 × (-3.321928) ≈ 144 × 3.321928 ≈ 478.36. That seems even more unreasonable. So the most plausible is log base 10.But maybe I should check the standard learning curve formula. Wait, the learning curve concept in psychology and education usually models the reduction in time or effort required as a function of practice, often following a power law. But the formula given here is t = -144 × log(1 - N/100). Let me see.If N is the typing speed, then 1 - N/100 would approach 0 as N approaches 100. So as N approaches 100, log(1 - N/100) approaches log(0), which is negative infinity, but multiplied by -144 gives positive infinity. So the time required to reach 100 characters per minute would be infinite, which might make sense as an asymptote. So at N=90, which is 90% of the maximum speed (100), the time required is 144 hours. That seems plausible. If the formula is t = -144 × log10(1 - N/100), then yes, plugging in N=90 gives 144 hours.Wait, but let's do the calculation again step by step to be sure.Given N = 90,First, compute 1 - N/100: 1 - 90/100 = 1 - 0.9 = 0.1Then take the logarithm: log10(0.1) = -1Multiply by -144: -144 × (-1) = 144So t = 144 hours.Therefore, the answer should be 144. But just to double-check, maybe there's a mistake in interpreting the formula. For instance, is the formula correctly written? The problem says "t(h) represents the learning time required to reach the typing level N". So t is a function of h? Wait, no, maybe that was a typo. Wait, the original problem states:"the function t = -144 × lg (1- N/100) representing the 'learning curve' of typing practice, where N represents typing speed (in characters per minute), and t(h) represents the learning time required to reach the typing level N"Wait, maybe there's a confusion here. Is t a function of h (hours), or is t directly the learning time? The notation t(h) usually means t is a function of h, but here the problem says t(h) represents the learning time required to reach N. But in the equation given, t is expressed in terms of N. So maybe it's a typo, and it should be t(N), meaning t is a function of N. So perhaps h is not a variable here. The problem might have meant that t is the time in hours, so t(N) = -144 × log(1 - N/100). So to get t when N is 90, substitute N=90 into the equation.Therefore, steps as before:1 - 90/100 = 0.1log10(0.1) = -1Multiply by -144: -144 * (-1) = 144.Therefore, t = 144 hours. That seems correct. Let me check once more.Alternatively, maybe the problem uses natural logarithm. Let's see. If log here is natural logarithm, then ln(0.1) ≈ -2.302585, so t = -144 * (-2.302585) ≈ 331.57 hours. But that's about 13.8 days of continuous practice, which seems a lot. Whereas 144 hours is 6 days of 24-hour practice, which is also a lot, but perhaps in terms of cumulative practice over a longer period.But the problem states "learning time required to reach a typing speed of 90 characters per minute". Maybe the formula is designed such that 100 characters per minute is the theoretical maximum, and approaching it requires more time. But 144 hours seems more reasonable for practice time. For example, if you practice 2 hours a day, that's 72 days, which is about 2.5 months. Whereas 331 hours would be about 5.5 months at 2 hours a day. It's possible, but unless the formula is specifically using natural logarithm, but given the notation "lg", which is commonly log base 10, I think 144 is the answer.Alternatively, maybe "lg" here refers to the binary logarithm, but that's less common. Let's check:lg(0.1) in base 2 is log2(0.1) ≈ -3.3219, so t = -144 * (-3.3219) ≈ 478.35 hours. That's even longer. So unless specified otherwise, "lg" is base 10.Therefore, the answer is 144 hours. So I think that's the correct answer.Wait, another thought. Maybe the original equation is t = -144 * ln(1 - N/100), but written as lg. If the problem is from a source where "lg" is used for natural logarithm, which is sometimes denoted as "ln". But in standard mathematics, "lg" is log base 10, "ln" is natural, "lb" or "ld" is log base 2. But maybe in Chinese contexts, "lg" might refer to natural logarithm? Hmm, not sure. But given that it's presented here in English, likely "lg" is log base 10.But to be absolutely thorough, let's check if maybe the formula is different. For example, in some learning curve models, the time decreases with practice, but here time increases with N, which makes sense because higher typing speed requires more practice. So as N increases, the argument inside the log, (1 - N/100), decreases, making the log more negative, and multiplied by -144, t becomes more positive. So for N approaching 100, t approaches infinity, which is an asymptote. That makes sense.Therefore, with N=90, the calculation is as before. Therefore, the answer is 144 hours.Final AnswerThe learning time required is boxed{144} hours.</think>Given the function ( t = -144 times lg left(1 - frac{N}{100}right) ), where ( N ) represents typing speed in characters per minute and ( t ) represents the learning time required in hours, we need to find the learning time required to reach a typing speed of 90 characters per minute.1. Substitute ( N = 90 ) into the function: [ t = -144 times lg left(1 - frac{90}{100}right) ]2. Calculate ( 1 - frac{90}{100} ): [ 1 - frac{90}{100} = 1 - 0.9 = 0.1 ]3. Compute the logarithm base 10 of 0.1: [ lg(0.1) = -1 ]4. Multiply by -144: [ t = -144 times (-1) = 144 ]Thus, the learning time required to reach a typing speed of 90 characters per minute is (boxed{144}) hours.,<think>Okay, let me try to solve this problem step by step. So, the function given is t = -144 × log(1 - N/100), where N is the typing speed in characters per minute, and t is the learning time required to reach that speed in hours. The question is asking for the time needed to reach 90 characters per minute. First, I need to make sure I understand the function correctly. The variable t is the time in hours, and N is the target typing speed. So, to find the time required to reach 90 characters per minute, I should substitute N = 90 into the equation and solve for t. Let me write that out. t = -144 × log(1 - 90/100)Simplify the expression inside the logarithm first. 90 divided by 100 is 0.9, so 1 - 0.9 is 0.1. So now the equation becomes:t = -144 × log(0.1)Wait, the logarithm here, is it base 10 or natural logarithm? The problem uses "lg" which typically stands for logarithm base 10. In some contexts, "lg" might mean base 2, but in mathematics and engineering, "lg" often refers to base 10, while "ln" is natural logarithm. Since the problem mentions "lg", I think it's safe to assume it's base 10. Let me confirm that. In Chinese contexts, "lg" is log base 10, so maybe in this problem as well. Assuming log base 10, then log(0.1) is log10(0.1). Since 0.1 is 10 to the power of -1, log10(0.1) is -1. Therefore, log(0.1) = -1.So substituting back into the equation:t = -144 × (-1) = 144.Wait, so that would mean t = 144 hours? That seems straightforward, but let me check again to make sure I didn't skip any steps or make a mistake.Starting again:Given t = -144 × log(1 - N/100). When N = 90:1 - 90/100 = 1 - 0.9 = 0.1So log(0.1) base 10 is indeed -1 because 10^-1 = 0.1.Therefore, t = -144 × (-1) = 144. So 144 hours.Is there a possibility that "lg" here refers to natural logarithm? If that's the case, then log(0.1) would be ln(0.1), which is approximately -2.302585. Then t would be -144 multiplied by that, which is approximately 144 × 2.302585 ≈ 331.57 hours. But that contradicts the problem's notation. The problem says "lg", so unless specified otherwise, it's base 10. Also, in some contexts, "lg" could be log base 2, but log base 2 of 0.1 is approximately -3.3219, leading to t ≈ 144 × 3.3219 ≈ 478.35 hours. That seems way too high. Given the problem uses "lg", which is commonly base 10, and the answer comes out to 144, which is a clean number, I think that's the intended answer. Let me check if substituting N = 90 gives 144, which makes sense. Wait, maybe the problem has a typo, or maybe I misread the function. Let me double-check the original equation. The user wrote: t = -144 × lg(1 - N/100). Yes, that's correct. So substituting N = 90 gives t = -144 × lg(0.1). Since lg(0.1) is -1, the negatives cancel, giving 144. That seems correct. Alternatively, let's test with another value. Suppose N = 0. Then t = -144 × lg(1 - 0) = -144 × lg(1) = -144 × 0 = 0. That makes sense; if you want to reach 0 characters per minute, you need 0 hours of practice. If N approaches 100, then 1 - N/100 approaches 0, so log(1 - N/100) approaches negative infinity, so t approaches positive infinity. Which also makes sense: to reach 100 characters per minute, you need infinite time, which is a common learning curve model where you approach an asymptote. So the model seems reasonable. Therefore, substituting N = 90, t = 144 hours. That seems correct. I don't see any mistakes in the reasoning. The answer should be 144.Final AnswerThe learning time required to reach a typing speed of 90 characters per minute is boxed{144} hours.</think>Given the function ( t = -144 times lg left(1 - frac{N}{100}right) ), where ( N ) represents typing speed in characters per minute and ( t ) represents the learning time in hours required to reach that speed, we need to find the time required to reach a typing speed of 90 characters per minute.1. Substitute ( N = 90 ) into the function: [ t = -144 times lg left(1 - frac{90}{100}right) ]2. Simplify the expression inside the logarithm: [ 1 - frac{90}{100} = 1 - 0.9 = 0.1 ]3. Calculate the logarithm (assuming base 10, as indicated by "lg"): [ lg(0.1) = -1 ]4. Substitute the logarithm value back into the equation: [ t = -144 times (-1) = 144 ]Thus, the learning time required to reach a typing speed of 90 characters per minute is (boxed{144}) hours.]

🔑:Since the function is t=-144timeslg (1- frac {N}{100}),When N=90 characters per minute,t=-144timeslg (1- frac {90}{100})=144.Therefore, the answer is boxed{144}.Substituting N=90 characters per minute into the function t=-144timeslg (1- frac {N}{100}) can yield the result.This problem tests the method of finding function values and is a basic question. When solving the problem, carefully read the question and pay attention to the reasonable application of function properties.

🧠:[<think>Okay, so I need to solve this system of inequalities. Let me take a deep breath and start step by step. There are two inequalities here:First one: 4x - 2 ≥ 3(x - 1)Second one: (x - 5)/2 > x - 4I need to solve each inequality separately first and then find the intersection of the solutions, right? Because it's a system, so both conditions have to be true at the same time. Alright, let's start with the first inequality.First inequality: 4x - 2 ≥ 3(x - 1)Hmm, okay. Let me distribute the 3 on the right side. So, 3 times (x - 1) is 3x - 3. So the inequality becomes:4x - 2 ≥ 3x - 3Now, I need to get all the x terms on one side and constants on the other. Let me subtract 3x from both sides. That would give:4x - 3x - 2 ≥ -3Which simplifies to:x - 2 ≥ -3Next, add 2 to both sides to isolate x:x ≥ -3 + 2Which is:x ≥ -1Okay, so the solution to the first inequality is x ≥ -1. Let me note that down.Now moving on to the second inequality: (x - 5)/2 > x - 4This is a bit trickier. Let me see. Maybe I can multiply both sides by 2 to eliminate the denominator. But first, I need to make sure that multiplying by 2 won't affect the inequality sign. Since 2 is positive, the inequality direction remains the same. So:Multiply both sides by 2:(x - 5) > 2(x - 4)Let me expand the right side:x - 5 > 2x - 8Now, let's get all x terms on one side. Subtract 2x from both sides:x - 2x - 5 > -8Which simplifies to:- x - 5 > -8Now, add 5 to both sides:- x > -8 + 5Which is:- x > -3Hmm, now I need to solve for x. If I multiply both sides by -1, that will reverse the inequality sign. So:x < 3Wait, let me check that again. If we have -x > -3, then multiplying both sides by -1 gives x < 3. Yep, that's correct. Because multiplying by a negative number flips the inequality.So the solution to the second inequality is x < 3.Now, putting both solutions together. The first inequality requires x ≥ -1 and the second requires x < 3. So the solution set is all real numbers x such that x is greater than or equal to -1 and less than 3.In interval notation, that would be [-1, 3). But since the problem is presented as a system, we need to present the solution in terms of x. So the answer is x ∈ [-1, 3).Wait a second, let me verify both inequalities with a test value in that interval to make sure. Let's pick x = 0, which is in the interval.First inequality: 4(0) - 2 ≥ 3(0 - 1) => -2 ≥ -3, which is true.Second inequality: (0 - 5)/2 > 0 - 4 => (-5)/2 > -4 => -2.5 > -4, which is also true.Another test value: x = -1, which is the lower bound.First inequality: 4(-1) - 2 ≥ 3(-1 - 1) => -4 -2 ≥ 3(-2) => -6 ≥ -6. That's true because it's equal.Second inequality: (-1 - 5)/2 > -1 -4 => (-6)/2 > -5 => -3 > -5. True as well.Now test x = 3, which is not included. Let's check the inequalities:First inequality: 4(3) - 2 ≥ 3(3 -1) => 12 -2 ≥ 3*2 => 10 ≥ 6, which is true.Second inequality: (3 -5)/2 > 3 -4 => (-2)/2 > -1 => -1 > -1. That's false because it's equal. So x=3 is not part of the solution, which matches our interval.What about a number less than -1, like x = -2.First inequality: 4(-2) -2 ≥ 3(-2 -1) => -8 -2 ≥ 3*(-3) => -10 ≥ -9, which is false. So that's correct, x can't be less than -1.And a number greater than 3, like x=4.First inequality: 4(4)-2 ≥ 3(4-1) => 16-2 ≥ 9 => 14 ≥9, true.Second inequality: (4 -5)/2 >4 -4 => (-1)/2 >0 => -0.5 >0, which is false. So x=4 is excluded.Therefore, the solution is indeed all real numbers from -1 to 3, including -1 but not including 3.So the final answer is x between -1 and 3, written as -1 ≤ x < 3.Final AnswerThe solution to the inequality system is boxed{[-1, 3)}.</think>To solve the system of inequalities:[left{begin{array}{l}4x - 2 ge 3(x - 1) frac{x - 5}{2} > x - 4end{array}right.]we start by solving each inequality separately.First inequality: (4x - 2 ge 3(x - 1))1. Distribute the 3 on the right side: [ 4x - 2 ge 3x - 3 ]2. Subtract (3x) from both sides: [ x - 2 ge -3 ]3. Add 2 to both sides: [ x ge -1 ]The solution to the first inequality is (x ge -1).Second inequality: (frac{x - 5}{2} > x - 4)1. Multiply both sides by 2 to clear the denominator: [ x - 5 > 2(x - 4) ]2. Expand the right side: [ x - 5 > 2x - 8 ]3. Subtract (2x) from both sides: [ -x - 5 > -8 ]4. Add 5 to both sides: [ -x > -3 ]5. Multiply by -1 and reverse the inequality: [ x < 3 ]The solution to the second inequality is (x < 3).Combining the solutions:The first inequality gives (x ge -1) and the second inequality gives (x < 3). The intersection of these solutions is:[-1 le x < 3]Thus, the solution to the inequality system is (boxed{[-1, 3)}).,<think>Okay, so I need to solve this system of inequalities. Let me see. The system is:[begin{cases}4x - 2 ge 3(x - 1) dfrac{x - 5}{2} > x - 4end{cases}]Alright, so I need to solve each inequality separately first and then find the intersection of the solutions, right? Because both inequalities have to be true at the same time for the system. Let me start with the first one.First inequality: 4x - 2 ≥ 3(x - 1)Hmm, let me expand the right side first. Distribute the 3 into (x - 1). So that would be 3*x - 3*1, which is 3x - 3. So the inequality becomes:4x - 2 ≥ 3x - 3Now, I need to get all the x terms on one side and constants on the other. Let me subtract 3x from both sides to get rid of the 3x on the right.4x - 3x - 2 ≥ -3That simplifies to:x - 2 ≥ -3Next, I can add 2 to both sides to isolate x.x - 2 + 2 ≥ -3 + 2Which simplifies to:x ≥ -1Okay, so the solution to the first inequality is all real numbers x such that x is greater than or equal to -1. Got that.Now onto the second inequality: (x - 5)/2 > x - 4Hmm, fractions can be tricky. Maybe I can eliminate the denominator by multiplying both sides by 2. Let me try that.Multiply both sides by 2:2*( (x - 5)/2 ) > 2*(x - 4)Simplify left side: (x - 5), and right side: 2x - 8So the inequality becomes:x - 5 > 2x - 8Now, let's get all x terms on one side. Subtract 2x from both sides:x - 2x - 5 > -8Which simplifies to:- x - 5 > -8Now, add 5 to both sides:- x > -8 + 5Which is:- x > -3Hmm, now I need to solve for x. When I multiply or divide by a negative number, the inequality sign flips. So, if I multiply both sides by -1, I need to reverse the inequality.So:x < 3Wait, let me check that again. If -x > -3, then multiplying both sides by -1 gives x < 3. Yes, that's right.So the solution to the second inequality is all real numbers x such that x is less than 3.Now, the system requires both inequalities to be true at the same time. So I need the intersection of x ≥ -1 and x < 3.So combining these two, x must be greater than or equal to -1 and less than 3. In interval notation, that would be [-1, 3). But let me represent it properly.So the solution set is all real numbers x where -1 ≤ x < 3.Wait, let me verify if I did everything correctly. Sometimes when dealing with inequalities, it's easy to make a mistake with the signs.Starting with the first inequality:4x - 2 ≥ 3(x - 1)4x - 2 ≥ 3x - 3Subtract 3x:4x - 3x -2 ≥ -3x -2 ≥ -3Add 2:x ≥ -1. Yes, that seems right.Second inequality:(x - 5)/2 > x - 4Multiply both sides by 2:x -5 > 2x -8Subtract 2x:- x -5 > -8Add 5:- x > -3Multiply by -1 (reverse inequality):x < 3. Correct.So combining both, x has to be between -1 (inclusive) and 3 (exclusive). So the solution is [-1, 3).But let me check with a number in that interval to see if both inequalities hold. Let's pick x = 0.First inequality: 4*0 -2 = -2. 3*(0 -1) = -3. So -2 ≥ -3? Yes, that's true.Second inequality: (0 -5)/2 = -5/2. 0 -4 = -4. So -2.5 > -4? Yes, that's true.Another test at x = -1.First inequality: 4*(-1) -2 = -4 -2 = -6. 3*(-1 -1) = 3*(-2) = -6. So -6 ≥ -6? Yes, equality holds, which is allowed.Second inequality: (-1 -5)/2 = -6/2 = -3. -1 -4 = -5. So -3 > -5? Yes, that's true.Now, test at x = 3 (which is excluded).First inequality: 4*3 -2 =12 -2=10. 3*(3 -1)=3*2=6. 10 ≥6? Yes, that's true.Second inequality: (3 -5)/2 = (-2)/2 = -1. 3 -4 = -1. So -1 > -1? No, because -1 is not greater than -1. So x=3 is not included in the solution, which matches the interval [-1,3).Another test at x=2.9.First inequality: 4*2.9 -2 =11.6 -2=9.6. 3*(2.9 -1)=3*1.9=5.7. 9.6 ≥5.7. True.Second inequality: (2.9 -5)/2= (-2.1)/2= -1.05. 2.9 -4= -1.1. So -1.05 > -1.1. Yes, because -1.05 is to the right of -1.1 on the number line.So that works. How about a number less than -1, say x = -2.First inequality: 4*(-2)-2 = -8-2=-10. 3*(-2 -1)=3*(-3)=-9. So -10 ≥ -9? No, which is correct since x=-2 is not in the solution.So the solution set is indeed all real numbers from -1 to 3, including -1 but not 3.Therefore, the answer should be written in interval notation as [-1, 3), or if they want it in set-builder notation, {x | -1 ≤ x < 3}. But since the problem says "put your final answer within boxed{}", which usually prefers interval notation, I think boxed answer would be boxed{[-1, 3)}.Final AnswerThe solution to the inequality system is boxed{[-1, 3)}.</think>To solve the system of inequalities:[begin{cases}4x - 2 ge 3(x - 1) dfrac{x - 5}{2} > x - 4end{cases}]we start by solving each inequality separately.First inequality: (4x - 2 ge 3(x - 1))1. Expand the right side: (4x - 2 ge 3x - 3)2. Subtract (3x) from both sides: (x - 2 ge -3)3. Add 2 to both sides: (x ge -1)Second inequality: (dfrac{x - 5}{2} > x - 4)1. Multiply both sides by 2: (x - 5 > 2x - 8)2. Subtract (2x) from both sides: (-x - 5 > -8)3. Add 5 to both sides: (-x > -3)4. Multiply by -1 and reverse the inequality: (x < 3)Combining the solutions:The first inequality gives (x ge -1) and the second inequality gives (x < 3). The intersection of these solutions is:[-1 le x < 3]Thus, the solution to the inequality system is (boxed{[-1, 3)}).]

🔑:To solve the given system of inequalities, we'll tackle each inequality one at a time.1. For the first inequality 4x - 2 geqslant 3(x - 1):begin{align*}4x - 2 &geqslant 3x - 3 4x - 3x &geqslant -3 + 2 x &geqslant -1end{align*}2. For the second inequality frac{x - 5}{2} > x - 4:begin{align*}frac{x - 5}{2} &> x - 4 x - 5 &> 2(x - 4) x - 5 &> 2x - 8 -5 + 8 &> 2x - x 3 &> xend{align*}Thus, x < 3.Combining the results from both inequalities, we find the solution set for the system is -1 leqslant x < 3. Therefore, the solution set of this inequality system is encapsulated as boxed{-1 leqslant x < 3}.

🧠:[<think>Okay, let me try to figure out this problem. So, we have the complex number (3i - 1)/(1 + 3i), and we need to find its coordinate in the complex plane. The options are A, B, C, D with different coordinates. Hmm, complex numbers in the plane are represented as (a, b) where the complex number is a + bi. So, basically, I need to simplify this fraction to the form a + bi and then the coordinate will be (a, b). That makes sense.Alright, how do I divide complex numbers? I remember that to divide complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of 1 + 3i is 1 - 3i. Let me try that.So, starting with (3i - 1)/(1 + 3i). Multiply numerator and denominator by (1 - 3i):[(3i - 1)(1 - 3i)] / [(1 + 3i)(1 - 3i)]First, let me compute the denominator. (1 + 3i)(1 - 3i) is a difference of squares: 1² - (3i)² = 1 - 9i². Since i² is -1, that becomes 1 - 9(-1) = 1 + 9 = 10. So the denominator is 10. That's straightforward.Now the numerator: (3i - 1)(1 - 3i). Let me expand this. Using distributive property:3i * 1 + 3i * (-3i) - 1 * 1 - 1 * (-3i)Calculate each term:First term: 3i * 1 = 3iSecond term: 3i * (-3i) = -9i²Third term: -1 * 1 = -1Fourth term: -1 * (-3i) = 3iCombine these terms:3i -9i² -1 +3iCombine like terms. The real parts are -9i² -1. Wait, but i² is -1, so -9i² is -9*(-1) = 9. Therefore, the real parts are 9 -1 = 8.The imaginary parts are 3i + 3i = 6i.So the numerator is 8 + 6i.Therefore, the entire expression becomes (8 + 6i)/10. Which simplifies to (8/10) + (6/10)i, which is 4/5 + 3/5i. So, in terms of coordinates, that's (4/5, 3/5). Let me check the options. Option A is (4/5, 3/5). So that's the answer?Wait, let me double-check my calculations because sometimes I might make a mistake in expanding.Original numerator: (3i -1)(1 -3i). Let's multiply it again:First, multiply -1 by 1: -1*1 = -1Then, -1 by -3i: +3iThen, 3i *1 = 3iThen, 3i * (-3i) = -9i²So adding up all these: -1 +3i +3i -9i²Combine like terms: (-1) + (3i +3i) + (-9i²) = -1 +6i -9i²Since i² = -1, -9i² = 9. So, -1 +6i +9 = 8 +6i. Yep, that's correct. Then divide by 10: 8/10 +6i/10 = 4/5 + 3i/5. So, coordinate (4/5, 3/5). Option A.But wait, let me check if the problem was written correctly. The original complex number is (3i -1)/(1 +3i). Let me confirm again. The numerator is 3i -1, which can also be written as -1 +3i. The denominator is 1 +3i. So we have (-1 +3i)/(1 +3i). Multiply numerator and denominator by (1 -3i):Numerator: (-1 +3i)(1 -3i) = -1*1 + (-1)*(-3i) +3i*1 +3i*(-3i) = -1 +3i +3i -9i² = -1 +6i +9 = 8 +6iDenominator: 10, as before. So yes, same result. So answer is A.But wait, wait. Let me check the options again. Option A is (4/5, 3/5). Option C is (3/5, 4/5). Did I mix up the real and imaginary parts? Wait, complex number is a + bi, so the coordinate is (a, b). In this case, 4/5 + 3/5 i, so (4/5, 3/5). That's correct. So answer is A.But let me just confirm once more. Maybe I made a mistake in the expansion. Let's try to compute the numerator again step by step.Numerator: (3i -1)(1 -3i)First term: 3i *1 = 3iSecond term: 3i*(-3i) = -9i² = 9 (since i² is -1)Third term: -1*1 = -1Fourth term: -1*(-3i) = 3iAdding all terms: 3i +9 -1 +3i = (9 -1) + (3i +3i) = 8 +6i. So yes, correct. Then divide by 10: 8/10 +6i/10 = 4/5 +3i/5. So coordinate is (4/5, 3/5). So the answer is A. Therefore, I think the correct answer is A.But wait, maybe I messed up the conjugation? Wait, no. The denominator is 1 +3i, so its conjugate is 1 -3i. Correct.Alternatively, maybe there's another way to approach this problem? Let me think. For example, recognizing that (3i -1) is similar to the denominator (1 +3i). Let me write both numerator and denominator:Numerator: -1 +3iDenominator: 1 +3iNotice that the numerator is -1 +3i, which is the negative of the denominator's conjugate. Let's see:Denominator's conjugate is 1 -3i. Then, negative of that is -1 +3i, which is exactly the numerator. So, numerator is - (conjugate of denominator). Therefore, (- conjugate(denominator))/denominator.So, if we let z = denominator = 1 +3i, then the expression is (- z̄)/z. Let's compute that:(- z̄)/z = - (z̄)/z = - ( (1 -3i)/(1 +3i) )But z/z̄ = (1 +3i)/(1 -3i) = [ (1 +3i)^2 ] / (1 +9) ) Hmm, maybe not helpful.Alternatively, since z = 1 +3i, then ( - z̄ ) / z = - (1 -3i)/(1 +3i). Let me compute this:Multiply numerator and denominator by (1 -3i):[ - (1 -3i) ] * (1 -3i) / [ (1 +3i)(1 -3i) ] = [ - (1 -3i)^2 ] / 10Compute (1 -3i)^2: 1 -6i +9i² = 1 -6i -9 = -8 -6i. So negative of that is 8 +6i. Then divide by 10: 8 +6i over 10. Same result as before. So, same answer. So (4/5, 3/5). Option A.Alternatively, maybe using polar form? Let's see. If I can convert numerator and denominator to polar form, then divide them. Let me try.First, compute the modulus and argument of numerator and denominator.Numerator: -1 +3i. The modulus is sqrt((-1)^2 +3^2) = sqrt(1 +9) = sqrt(10). The argument is arctangent of (3/-1). Since the point is in the second quadrant (real part negative, imaginary part positive), the argument is π - arctan(3/1) = π - arctan(3).Denominator: 1 +3i. The modulus is sqrt(1^2 +3^2) = sqrt(10). The argument is arctan(3/1) = arctan(3).Therefore, when we divide two complex numbers in polar form, we divide their moduli and subtract their arguments.So, modulus of result is sqrt(10)/sqrt(10) = 1. The argument is (π - arctan(3)) - arctan(3) = π - 2 arctan(3).But then, converting back to rectangular form: cos(π - 2 arctan(3)) + i sin(π - 2 arctan(3)).Hmm, maybe this is more complicated. Let me compute 2 arctan(3). Let's set θ = arctan(3). Then tanθ = 3. Then, we need to compute cos(π - 2θ) and sin(π - 2θ).Using trigonometric identities:cos(π - 2θ) = -cos(2θ)sin(π - 2θ) = sin(2θ)So, let's compute cos(2θ) and sin(2θ). Since tanθ = 3, we can imagine a right triangle with opposite side 3, adjacent side 1, hypotenuse sqrt(10). Therefore:sinθ = 3/sqrt(10)cosθ = 1/sqrt(10)Then,cos(2θ) = cos²θ - sin²θ = (1/10 - 9/10) = -8/10 = -4/5sin(2θ) = 2 sinθ cosθ = 2*(3/sqrt(10))*(1/sqrt(10)) = 6/10 = 3/5Therefore, cos(π - 2θ) = -cos(2θ) = -(-4/5) = 4/5sin(π - 2θ) = sin(2θ) = 3/5Therefore, the complex number is 4/5 + (3/5)i, which gives the coordinate (4/5, 3/5). So again, the answer is A. So this method also leads to the same result. Therefore, I can be confident that Option A is correct.Wait, but the problem gives the options and option A is (4/5, 3/5). But let me check once again, perhaps in case I made a mistake in the problem statement.Wait, the problem says: the coordinate of the point corresponding to the complex number (3i -1)/(1 +3i). So yes, that's the same as (-1 +3i)/(1 +3i). So we did that.Alternatively, maybe the problem was mistyped? But as per the user's message, it's (3i -1)/(1 +3i). So that's correct. Then, after simplifying, we get 4/5 +3/5i. So coordinate (4/5, 3/5). So answer A.But let me check why other options are there. For example, option C is (3/5, 4/5). Maybe someone might have flipped the real and imaginary parts. But in our calculation, the real part is 4/5 and the imaginary part is 3/5, so the coordinate is (4/5, 3/5). Therefore, A is correct.Alternatively, if someone miscalculates and thinks that the real part is 3/5 and imaginary part is 4/5, they might choose C. But in our case, it's A.Therefore, after verifying through two different methods (multiplying by conjugate and polar form), I can confirm that the correct answer is A.Final Answerboxed{A}</think>To determine the coordinate of the point corresponding to the complex number (frac{3i - 1}{1 + 3i}) in the complex plane, we need to simplify the given complex number to the form (a + bi).First, we multiply the numerator and the denominator by the conjugate of the denominator, which is (1 - 3i):[frac{3i - 1}{1 + 3i} cdot frac{1 - 3i}{1 - 3i} = frac{(3i - 1)(1 - 3i)}{(1 + 3i)(1 - 3i)}]Calculating the denominator:[(1 + 3i)(1 - 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10]Expanding the numerator:[(3i - 1)(1 - 3i) = 3i cdot 1 + 3i cdot (-3i) - 1 cdot 1 - 1 cdot (-3i)][= 3i - 9i^2 - 1 + 3i][= 3i - 9(-1) - 1 + 3i][= 3i + 9 - 1 + 3i][= 8 + 6i]Thus, the simplified complex number is:[frac{8 + 6i}{10} = frac{8}{10} + frac{6}{10}i = frac{4}{5} + frac{3}{5}i]The coordinate corresponding to the complex number (frac{4}{5} + frac{3}{5}i) is (left( frac{4}{5}, frac{3}{5} right)).Therefore, the correct answer is (boxed{A}).,<think>Okay, so I need to find the coordinate of the point in the complex plane that corresponds to the complex number (3i - 1)/(1 + 3i). The options are given as A, B, C, D. Let me think through this step by step.First, I remember that in the complex plane, a complex number a + bi is represented as the point (a, b). So, if I can simplify the given complex number (3i - 1)/(1 + 3i) to the standard form a + bi, then I can directly read off the coordinates (a, b) and match it with the options.To simplify (3i - 1)/(1 + 3i), I should multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 1 + 3i is 1 - 3i. Multiplying both numerator and denominator by this should eliminate the imaginary part in the denominator.Let me write that out:Start with:( -1 + 3i ) / (1 + 3i )Multiply numerator and denominator by (1 - 3i):[ (-1 + 3i)(1 - 3i) ] / [ (1 + 3i)(1 - 3i) ]First, let me compute the denominator. Using the formula (a + b)(a - b) = a² - b², so:Denominator = (1)^2 - (3i)^2 = 1 - 9i². Since i² = -1, this becomes 1 - 9(-1) = 1 + 9 = 10. So the denominator is 10.Now the numerator: (-1)(1) + (-1)(-3i) + (3i)(1) + (3i)(-3i)Compute each term:First term: -1 * 1 = -1Second term: -1 * -3i = 3iThird term: 3i * 1 = 3iFourth term: 3i * -3i = -9i²Combine these terms:-1 + 3i + 3i -9i²Combine like terms:-1 + 6i -9i²Again, since i² = -1, then -9i² = -9*(-1) = 9. So substitute that in:-1 + 6i + 9 = ( -1 + 9 ) + 6i = 8 + 6iTherefore, the numerator is 8 + 6i, and the denominator is 10. So the entire expression becomes:(8 + 6i)/10 = (8/10) + (6/10)i = (4/5) + (3/5)iSo in standard form a + bi, this is 4/5 + 3/5 i. Therefore, the corresponding point in the complex plane is (4/5, 3/5). Let me check the options:A: (4/5, 3/5) – that's exactly what I got.B: (-1, 3/5) – that's different.C: (3/5, 4/5) – swapped coordinates.D: (3/5, 1) – also different.So the answer must be option A. But wait, let me verify once more to be sure I didn't make any calculation errors.Wait, let me double-check the numerator computation:(-1 + 3i)(1 - 3i)Using distributive property:-1*1 = -1-1*(-3i) = 3i3i*1 = 3i3i*(-3i) = -9i² = +9So adding those up: -1 + 3i + 3i +9 = ( -1 +9 ) + (3i +3i ) = 8 +6i. That's correct. Then divide by 10: 8/10 + 6i/10 = 4/5 + 3/5i. So yes, (4/5, 3/5) is correct, which is option A.But wait, hold on. Let me check if the original problem was written correctly. The numerator is (3i -1), which is the same as (-1 + 3i). The denominator is (1 + 3i). So everything seems okay.Alternatively, maybe there's another way to approach this problem? For example, by recognizing that both numerator and denominator might be related by a factor. Let's see:Let me write both numerator and denominator:Numerator: -1 + 3iDenominator: 1 + 3iNotice that if I factor out a -1 from the numerator: -1*(1 - 3i). So then:Numerator becomes -1*(1 - 3i)Denominator is (1 + 3i)So the expression becomes:[-1*(1 - 3i)] / (1 + 3i) = -1 * (1 - 3i)/(1 + 3i)Now, if I let z = (1 - 3i)/(1 + 3i), then maybe compute z first.Compute z = (1 - 3i)/(1 + 3i). Multiply numerator and denominator by (1 - 3i) again? Wait, but that would be the conjugate. Wait, actually, in the previous approach, the denominator became 10. Wait, but if I do this:z = (1 - 3i)/(1 + 3i) => Multiply numerator and denominator by (1 - 3i):[(1 - 3i)^2]/[(1)^2 - (3i)^2] = (1 - 6i + 9i²)/(1 - 9i²) = (1 -6i -9)/(1 +9) = (-8 -6i)/10 = (-4 -3i)/5So z = (-4 -3i)/5Therefore, the original expression is -1 * z = -1 * (-4 -3i)/5 = (4 +3i)/5 = 4/5 + 3/5 i. So same result. So coordinates (4/5, 3/5), which is option A. So this confirms the answer.Alternatively, another approach: Recognize that multiplying numerator and denominator by i. Let's see:Original expression: ( -1 + 3i ) / (1 + 3i )Multiply numerator and denominator by i:Numerator: (-1 + 3i)*i = -i + 3i^2 = -i -3Denominator: (1 + 3i)*i = i + 3i^2 = i -3So the expression becomes ( -3 -i ) / ( -3 + i )Hmm, not sure if this helps. Let me see:( -3 -i ) / ( -3 + i )Multiply numerator and denominator by the conjugate of denominator, which is -3 -i:Numerator: (-3 -i)(-3 -i) = 9 +6i +i² = 9 +6i -1 = 8 +6iDenominator: (-3 +i)(-3 -i) = 9 +3i -3i -i² = 9 - (-1) = 10So again, (8 +6i)/10 = 4/5 + 3/5i. So same result. Hence, Option A.Therefore, I can be confident that Option A is the correct answer.Final Answerboxed{A}</think>To find the coordinate of the point corresponding to the complex number (frac{3i - 1}{1 + 3i}) in the complex plane, we need to simplify the given complex number to the standard form (a + bi).First, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is (1 - 3i):[frac{-1 + 3i}{1 + 3i} times frac{1 - 3i}{1 - 3i}]Calculating the denominator:[(1 + 3i)(1 - 3i) = 1^2 - (3i)^2 = 1 - 9i^2 = 1 + 9 = 10]Calculating the numerator:[(-1 + 3i)(1 - 3i) = -1 cdot 1 + (-1) cdot (-3i) + 3i cdot 1 + 3i cdot (-3i)][= -1 + 3i + 3i - 9i^2][= -1 + 6i - 9(-1)][= -1 + 6i + 9][= 8 + 6i]Thus, the simplified form of the complex number is:[frac{8 + 6i}{10} = frac{4}{5} + frac{3}{5}i]The corresponding coordinate in the complex plane is (left( frac{4}{5}, frac{3}{5} right)), which corresponds to option A.[boxed{A}]]

🔑:Since frac {3i-1}{1+3i}= frac {(3i-1)(1-3i)}{(1+3i)(1-3i)}= frac {8+6i}{10}= frac {4}{5}+ frac {3}{5}i, the coordinate of the point corresponding to the complex number frac {3i-1}{1+3i} is left( frac {4}{5}, frac {3}{5}right). Therefore, the correct answer is boxed{A}. This solution directly uses the algebraic operations of multiplication and division of complex numbers in algebraic form to simplify and find the answer. This question tests the algebraic operations of multiplication and division of complex numbers, the algebraic representation of complex numbers, and their geometric meaning, making it a basic question.

🧠:[<think>Okay, so I need to solve for x squared minus y squared given these two equations: (x + y)/2 = 5 and (x - y)/2 = 2. Hmm, let me think. Alright, first, maybe I can simplify these equations to find x and y individually. Let me write them down again to make sure I have them right.First equation: (x + y)/2 = 5. Second equation: (x - y)/2 = 2. So, if I multiply both sides of each equation by 2, that should get rid of the denominators. Let's try that.For the first equation: (x + y)/2 = 5. Multiplying both sides by 2 gives x + y = 10. Okay, that seems straightforward. Similarly, for the second equation: (x - y)/2 = 2. Multiplying both sides by 2 gives x - y = 4. Alright, now I have two equations:1. x + y = 102. x - y = 4Hmm, these look like a system of linear equations. I can solve them using either substitution or elimination. Let me try elimination here because the coefficients of y are +1 and -1, which would cancel out if I add the equations. Let's add equation 1 and equation 2 together.Adding the left sides: (x + y) + (x - y) = x + y + x - y = 2x. Adding the right sides: 10 + 4 = 14. So, 2x = 14. Dividing both sides by 2 gives x = 7. Okay, so x is 7. Now, to find y, I can substitute x back into one of the original equations. Let's take the first equation: x + y = 10. If x is 7, then 7 + y = 10. Subtracting 7 from both sides gives y = 3. Let me check that with the second equation to make sure. If x is 7 and y is 3, then x - y = 7 - 3 = 4, which matches the second equation. Great, so x = 7 and y = 3.Now, the question is asking for x² - y². Hmm, I remember that x squared minus y squared is a difference of squares, which can be factored as (x - y)(x + y). Let me verify that: (x - y)(x + y) = x(x + y) - y(x + y) = x² + xy - xy - y² = x² - y². Yeah, that works. So instead of calculating x squared and y squared separately and then subtracting, which might take longer, I can just use this factored form.From the equations we had earlier, x + y = 10 and x - y = 4. So, x² - y² = (x - y)(x + y) = 4 * 10 = 40. Wait, that's much simpler! So the answer should be 40. Let me just double-check by calculating x squared and y squared individually. If x is 7, then x squared is 49, and y is 3, so y squared is 9. Then 49 - 9 is indeed 40. Perfect, that confirms it. So regardless of the method, the answer is 40.But wait, just to make sure I didn't skip any steps or make any miscalculations. Let's recap:1. Original equations: (x + y)/2 = 5 and (x - y)/2 = 2.2. Multiply both equations by 2: x + y = 10 and x - y = 4.3. Add the two equations: 2x = 14 ⇒ x = 7.4. Substitute x = 7 into x + y = 10: y = 3.5. Therefore, x² - y² = (7)² - (3)² = 49 - 9 = 40.6. Alternatively, factor as (x - y)(x + y) = 4 * 10 = 40.Everything checks out. I don't think I made any mistakes here. The key was recognizing that x squared minus y squared can be factored into (x - y)(x + y), which allows us to use the values we already have from the original equations. That saved time instead of computing each square separately. But even if I had done it the longer way, I still end up with the same answer, so that's good.Another way to think about it: since they gave me equations for (x + y)/2 and (x - y)/2, maybe I could have noticed that x + y and x - y are directly related to those. But since they were divided by 2, multiplying by 2 was necessary first. So the process of solving for x and y was straightforward, and once you have x and y, plugging them into the expression is easy. But factoring x² - y² is a nice shortcut here. It's good that they taught us that identity in algebra, it comes in handy.So, yeah, all steps seem correct. I don't see any errors in the reasoning. The answer should be 40.Final AnswerThe value of (x^2 - y^2) is boxed{40}.</think>Given the equations (frac{x+y}{2} = 5) and (frac{x-y}{2} = 2), we start by eliminating the denominators by multiplying both sides of each equation by 2:[begin{align*}frac{x+y}{2} &= 5 implies x + y = 10, frac{x-y}{2} &= 2 implies x - y = 4.end{align*}]We now have the system of linear equations:1. (x + y = 10)2. (x - y = 4)Adding these two equations, we get:[(x + y) + (x - y) = 10 + 4 implies 2x = 14 implies x = 7.]Substituting (x = 7) back into the first equation (x + y = 10):[7 + y = 10 implies y = 3.]To find (x^2 - y^2), we use the difference of squares factorization:[x^2 - y^2 = (x - y)(x + y).]Substituting the values of (x - y = 4) and (x + y = 10):[x^2 - y^2 = 4 times 10 = 40.]Thus, the value of (x^2 - y^2) is (boxed{40}).,<think>Okay, so I need to solve for x squared minus y squared given these two equations: (x + y)/2 = 5 and (x - y)/2 = 2. Hmm, let me think. First, maybe I can simplify these equations to find x and y individually. Starting with the first equation: (x + y)/2 = 5. If I multiply both sides by 2, that should eliminate the denominator. So, 2*(x + y)/2 = 5*2. That simplifies to x + y = 10. Okay, got that. Then the second equation is (x - y)/2 = 2. Similarly, I can multiply both sides by 2. So, 2*(x - y)/2 = 2*2, which simplifies to x - y = 4. Nice, so now we have two simpler equations: x + y = 10 and x - y = 4. Now, I can solve these two equations simultaneously. Maybe adding them together? Let me try that. If I add the left sides: (x + y) + (x - y) = x + y + x - y = 2x. And the right sides: 10 + 4 = 14. So, 2x = 14. Dividing both sides by 2 gives x = 7. Once I have x, I can substitute back into one of the original equations to find y. Let's use the first equation: x + y = 10. If x is 7, then 7 + y = 10. Subtract 7 from both sides: y = 3. So, x is 7 and y is 3. Therefore, x squared minus y squared would be 7 squared minus 3 squared. Calculating that: 49 - 9 = 40. Wait, but hold on a second. I remember there's a formula for the difference of squares: x² - y² = (x + y)(x - y). Maybe I can use that instead of calculating each square separately. Let me check. From the original simplified equations, x + y = 10 and x - y = 4. If I multiply those two together: 10 * 4 = 40. That's the same result as before! So, whether I compute each square and subtract or use the difference of squares formula, I get the same answer. That's a good consistency check. So, the value of x² - y² is 40. I think that's the answer. Let me just verify the initial equations with x=7 and y=3 to make sure I didn't make a mistake. First equation: (7 + 3)/2 = 10/2 = 5, which matches. Second equation: (7 - 3)/2 = 4/2 = 2, which also matches. Perfect, so the values of x and y are correct. Therefore, the answer should be 40. I don't see any mistakes in the steps. I used two different methods to compute x² - y², both giving the same result, and verified the solutions in the original equations. Seems solid.Final AnswerThe value of (x^2 - y^2) is boxed{40}.</think>Given the equations (frac{x+y}{2} = 5) and (frac{x-y}{2} = 2), we start by simplifying them. First, multiplying both sides of (frac{x+y}{2} = 5) by 2 gives:[x + y = 10]Next, multiplying both sides of (frac{x-y}{2} = 2) by 2 gives:[x - y = 4]We now have the system of equations:[begin{cases}x + y = 10 x - y = 4end{cases}]Adding these two equations together:[(x + y) + (x - y) = 10 + 4 implies 2x = 14 implies x = 7]Substituting (x = 7) back into (x + y = 10):[7 + y = 10 implies y = 3]To find (x^2 - y^2), we use the difference of squares formula:[x^2 - y^2 = (x + y)(x - y)]Substituting the known values (x + y = 10) and (x - y = 4):[x^2 - y^2 = 10 times 4 = 40]Thus, the value of (x^2 - y^2) is (boxed{40}).]

🔑:Solution 1Since frac{x+y}{2}=5, then x+y=2 cdot 5=10. Since frac{x-y}{2}=2, then x-y=2 cdot 2=4.Therefore, x^{2}-y^{2}=(x+y)(x-y)=10 cdot 4=40.## Solution 2Since frac{x+y}{2}=5 and frac{x-y}{2}=2, then frac{x+y}{2}+frac{x-y}{2}=5+2 which simplifies to x=7.Also, frac{x+y}{2}-frac{x-y}{2}=5-2 which simplifies to y=3.Therefore, x^{2}-y^{2}=7^{2}-3^{2}=49-9=40.ANSWER: 40