🧠:[<think>Okay, let's try to figure out which number is the largest among P, Q, R, S, and T. All of them are different integers between 2 and 19 inclusive. Let's go through each condition one by one and see what possible numbers each could be.Starting with P: it's a two-digit prime number whose digits add up to a prime number. Since P is between 2 and 19, the two-digit primes in that range are 11, 13, 17, and 19. Now, let's check the sum of their digits:- 11: 1 + 1 = 2, which is prime.- 13: 1 + 3 = 4, not prime.- 17: 1 + 7 = 8, not prime.- 19: 1 + 9 = 10, not prime.So P must be 11. Wait, hold on, but 2 is a single-digit number. The problem says P is a two-digit prime, so 11 is the only one here where the sum of digits is prime. So P = 11? Let me confirm. The two-digit primes between 2 and 19 are indeed 11,13,17,19. But wait, 11 is two-digit, right? Yeah, 1 and 1. So digits sum to 2, which is prime. So P is definitely 11. Okay, so that's P.Next, Q is a multiple of 5. Numbers between 2 and 19 that are multiples of 5: 5,10,15. But since all numbers must be different and between 2 and 19 inclusive, Q could be 5,10,15. But wait, 5 is a single-digit number, but Q is a number between 2 and 19, so 5 is allowed. However, we need to check if these numbers are available because other numbers might take them. Let's note possible Q as 5,10,15.R is an odd number, but not a prime number. So odd composites between 2 and 19. Let's list them: 9,15. Because 9 is 3×3, 15 is 3×5. Are there more? 21 is too high. Wait, 25 is over 19. So R can be 9 or 15. But 15 is also a multiple of 5, which Q could be. So if Q is 15, R can't be 15. So possible R: 9 and 15, but need to check availability.S is the square of a prime number. Let's see. Prime numbers whose squares are between 2 and 19. Primes to consider: 2,3,5,7. Their squares are 4,9,25,49. 25 and 49 are too big. So squares are 4 and 9. 4 is 2², and 9 is 3². So S can be 4 or 9. But S has to be between 2 and 19, so 4 and 9. However, 9 is also a candidate for R. So if R is 9, then S can't be 9. So S is either 4 or 9. Let's note that.T is a prime number that is also the mean (average) of P and Q. Since P is 11, as we found earlier, then T = (P + Q)/2 = (11 + Q)/2. T must be a prime number, and also an integer, which means that (11 + Q) must be even, so Q must be odd. Because 11 is odd, odd + odd = even. So Q must be odd. Looking back at Q's possibilities: 5,10,15. If Q has to be odd, then Q can't be 10. So Q must be 5 or 15. So Q is either 5 or 15. Let's keep that in mind.So T must be (11 + Q)/2, and T must be a prime number. Let's check both possibilities for Q:1. If Q is 5:T = (11 + 5)/2 = 16/2 = 8. But 8 is not prime. So that's invalid. So Q can't be 5.2. If Q is 15:T = (11 + 15)/2 = 26/2 = 13. 13 is a prime number. So T = 13. That works. So Q must be 15, T must be 13.So now, let's confirm:P = 11 (two-digit prime, digits sum to 2, which is prime).Q = 15 (multiple of 5).T = 13 (prime number, average of 11 and 15).Now, moving on to R and S. R is an odd composite number. The candidates are 9 and 15. But Q is already 15, so R can't be 15. Therefore, R must be 9. Then S is the square of a prime number. The remaining options for S are 4 or 9. But R is 9, so S must be 4. Because 9 is already taken by R, so S = 4.So now, the numbers we have are:P = 11,Q = 15,R = 9,S = 4,T = 13.All of these are distinct integers between 2 and 19. Now, the question is: which is the largest? Let's list them:11,15,9,4,13. The largest is 15, which is Q. So the answer should be B) Q.Wait, let's double-check if there are any other possibilities. For example, if S was 9, then R would have to be 15. But Q is already 15, so that's not possible. So S must be 4 and R must be 9. That seems correct. Let's check if T is a prime. 13 is prime. Q is 15, which is allowed as a multiple of 5. P is 11. All numbers are distinct. So yes, the largest is Q = 15. So the answer is B.But wait, just to be thorough, let's check if there are other possibilities for P. Wait, earlier I concluded P must be 11. Let me check that again.Original P: two-digit prime between 2 and 19. So possible two-digit primes are 11,13,17,19. Then sum of digits:- 11: 1+1=2 (prime)- 13:1+3=4 (not prime)-17:1+7=8 (not prime)-19:1+9=10 (not prime)So only 11 satisfies that condition. So P must be 11. Therefore, that's fixed. So Q must be 15, leading to T=13, R=9, S=4. All numbers are unique, and within the range. Thus, Q=15 is the largest. So the answer is B.Final Answerboxed{B}</think>Given the problem, we need to determine the largest number among ( P, Q, R, S, ) and ( T ) based on the provided conditions. Here is the step-by-step reasoning:1. Condition for ( P ): - ( P ) is a two-digit prime number whose digits add up to a prime number. - Possible two-digit primes between 2 and 19: 11, 13, 17, 19. - Sum of digits: 11 (1+1=2, prime), 13 (1+3=4, not prime), 17 (1+7=8, not prime), 19 (1+9=10, not prime). - Therefore, ( P = 11 ).2. Condition for ( Q ): - ( Q ) is a multiple of 5. - Possible values: 5, 10, 15. - Since ( T ) (the mean of ( P ) and ( Q )) must be an integer and prime, ( Q ) must be odd (to make ( (11 + Q) ) even). - Thus, ( Q ) can be 5 or 15. Checking ( T ): - If ( Q = 5 ), ( T = frac{11 + 5}{2} = 8 ) (not prime). - If ( Q = 15 ), ( T = frac{11 + 15}{2} = 13 ) (prime). - Therefore, ( Q = 15 ) and ( T = 13 ).3. Condition for ( R ): - ( R ) is an odd number, but not a prime. - Possible values: 9, 15. - Since ( Q = 15 ), ( R ) must be 9.4. Condition for ( S ): - ( S ) is the square of a prime number. - Possible values: 4 (2²), 9 (3²). - Since ( R = 9 ), ( S ) must be 4.5. Verification: - All numbers ( P = 11 ), ( Q = 15 ), ( R = 9 ), ( S = 4 ), and ( T = 13 ) are distinct integers between 2 and 19. - The largest number is ( Q = 15 ).Thus, the answer is (boxed{B}).,<think>Okay, let's tackle this problem step by step. So we have five different integers between 2 and 19 inclusive: P, Q, R, S, and T. Each has different properties, and we need to figure out which one is the largest. The options are P, Q, R, S, or T. Let's break down each condition one by one and see what numbers they could be, then compare them.First, let's note that all numbers are between 2 and 19, inclusive, and they are all different. So we can't have duplicates, and each has to fit their specific criteria.Starting with P: it's a two-digit prime number whose digits add up to a prime number. Let's list all two-digit prime numbers between 2 and 19. Wait, two-digit primes between 2 and 19 would be from 10 to 19. But 10 isn't prime. Let's list them:11, 13, 17, 19. Those are the two-digit primes in that range. Now, we need to check which of these have digits that add up to a prime number.11: 1 + 1 = 2, which is prime. So 11 is a candidate.13: 1 + 3 = 4, which is not prime. So 13 is out.17: 1 + 7 = 8, not prime. So 17 is out.19: 1 + 9 = 10, which is not prime. So 19 is out.Wait, so the only possible P is 11? Hmm. Wait, let me check again. Wait, 11 is the only two-digit prime in 10-19 where digits add up to a prime. So P must be 11. Hmm. Let me confirm:Wait, 11: digits sum to 2 (prime). Correct.13: 1+3=4 (not prime). 17: 1+7=8 (not prime). 19: 1+9=10 (not prime). Yeah, so P is definitely 11. Okay, so P=11.Moving on to Q: Q is a multiple of 5. Numbers between 2 and 19 that are multiples of 5: 5, 10, 15. But since Q has to be between 2 and 19 inclusive, and different from P (which is 11). So possible Qs are 5, 10, 15. But wait, Q is between 2 and 19, so 5, 10, 15. Let's note these as possibilities: Q could be 5, 10, or 15. However, we need to keep in mind that all numbers must be different. Since P is 11, Q can't be 11. So Q is either 5, 10, or 15. Let's hold onto that for now.Next, R is an odd number, but not a prime number. So R is a composite odd number between 2 and 19. Let's list the odd numbers between 2 and 19 first:3, 5, 7, 9, 11, 13, 15, 17, 19.But R is not a prime, so we exclude primes. The primes in this list are: 3, 5, 7, 11, 13, 17, 19. So composite odd numbers would be 9, 15. Wait, 9 is 3×3, composite. 15 is 3×5, composite. So R could be 9 or 15. Also, is 15 included here? Yes. So possible R: 9, 15. However, since Q could be 15 (if Q is 15), but Q is a multiple of 5, which 15 is. So if Q is 15, then R can't be 15. So R is either 9 or 15, depending on Q's value.Then S is the square of a prime number. Let's list squares of primes between 2 and 19.Prime numbers first: primes between 2 and 19 are 2, 3, 5, 7, 11, 13, 17, 19. Their squares:2²=4, 3²=9, 5²=25 (too big, since 25>19), 7²=49 (too big), same for the rest. So the squares of primes in the range 2-19 are 4, 9. But 4 is 2², which is prime squared. 9 is 3². So S can be 4 or 9. However, note that S is between 2 and 19, so 4 and 9 are the options. But let's check if these numbers are available. If S is 4, but 4 is even, but maybe. If S is 9, but R could be 9. So if S is 9, then R can't be 9. Similarly, if S is 4, then that's a different number. So possible S: 4 or 9. Let's note that.Finally, T is a prime number that is also the mean (average) of P and Q. Remember that P is 11, so T = (11 + Q)/2. Since T must be a prime number and an integer. So (11 + Q) must be even, which means Q must be odd because 11 is odd. Odd plus odd is even. So Q must be odd. However, Q is a multiple of 5. The multiples of 5 between 2 and 19 are 5, 10, 15. Of these, the odd ones are 5 and 15. So Q must be either 5 or 15. Therefore, Q can't be 10, because 10 is even. So Q is either 5 or 15.Let's check both possibilities.First, if Q is 5:Then T = (11 + 5)/2 = 16/2 = 8. But 8 is not a prime number. So this is invalid. Therefore, Q cannot be 5. Therefore, Q must be 15.Let's check Q=15:T = (11 + 15)/2 = 26/2 = 13. 13 is a prime number. Perfect. So T=13, which is prime, and it's the average of P=11 and Q=15. So Q=15, T=13.Now, let's check the other variables.So far, we have P=11, Q=15, T=13.Now, R is an odd composite number. The candidates were 9 and 15. But Q is already 15, so R must be 9. So R=9.Then S is the square of a prime number. The candidates are 4 and 9. But R=9 is already taken, so S must be 4. Therefore, S=4.So now, all numbers: P=11, Q=15, R=9, S=4, T=13. All are distinct and between 2 and 19.Now, the question is which of these is the largest. Let's list them:P=11, Q=15, R=9, S=4, T=13.So Q=15 is the largest. Therefore, the answer is Q, which is option B.But wait, let me double-check all the constraints to make sure we didn't miss anything.All numbers are different: 11,15,9,4,13. Yes, all different.P: two-digit prime (11) with digits summing to prime (1+1=2). Check.Q: multiple of 5 (15). Check.R: odd composite (9). Check.S: square of prime (4=2²). Check.T: prime number (13) and average of P and Q (11 and 15 average to 13). Check.All numbers are between 2 and 19. Check.So Q=15 is indeed the largest. Therefore, the answer is B.Final Answerboxed{B}</think>Given the problem, we need to determine the largest number among (P, Q, R, S,) and (T) based on specific constraints. Here's the step-by-step reasoning:1. Identifying (P): - (P) is a two-digit prime number whose digits add up to a prime number. - Possible two-digit primes between 2 and 19: 11, 13, 17, 19. - Checking digits sum: 11 (1+1=2, prime), 13 (1+3=4, not prime), 17 (1+7=8, not prime), 19 (1+9=10, not prime). - Therefore, (P = 11).2. Identifying (Q): - (Q) is a multiple of 5. - Possible values: 5, 10, 15. - Since (T) (the mean of (P) and (Q)) must be a prime, (Q) must be odd (5 or 15). - Checking (Q = 5): (T = (11 + 5)/2 = 8) (not prime). - Checking (Q = 15): (T = (11 + 15)/2 = 13) (prime). - Therefore, (Q = 15) and (T = 13).3. Identifying (R): - (R) is an odd composite number. - Possible values: 9, 15. - Since (Q = 15), (R) must be 9. - Therefore, (R = 9).4. Identifying (S): - (S) is the square of a prime number. - Possible values: 4 (2²), 9 (3²). - Since (R = 9), (S) must be 4. - Therefore, (S = 4).5. Verification: - All numbers (P = 11), (Q = 15), (R = 9), (S = 4), and (T = 13) are distinct and within the range 2 to 19. - The largest number among these is (Q = 15).Thus, the largest number is (boxed{B}).]

🔑:Let us first consider the possibilities for each integer separately:- The two-digit prime numbers are 11,13,17,19. The only one whose digits add up to a prime number is 11 . Therefore, P=11.- Since Q is a multiple of 5 between 2 and 19, then the possible values of Q are 5,10,15.- The odd numbers between 2 and 19 that are not prime are 9 and 15 , so the possible values of R are 9 and 15 .- The squares between 2 and 19 are 4, 9 and 16 . Only 4 and 9 are squares of prime numbers, so the possible values of S are 4 and 9 .- Since P=11, the possible value of Q are 5,10 and 15, and T is the average of P and Q, then T could be 8,10.5 or 13 . Since T is also a prime number, then T must be 13, so Q=15.We now know that P=11, Q=15 and T=13.Since the five numbers are all different, then R cannot be 15 , so R=9.Since R=9, S cannot be 9 , so S=4.Therefore, the largest of the five integers is Q=15.ANsWER: (B)

🧠:[<think>Okay, let's see. I have this question here asking which of the four calculations is correct: A, B, C, or D. Each option involves exponents, so I need to recall the rules for handling exponents. Hmm, let's go through them one by one.Starting with option A: a squared multiplied by a cubed equals a to the sixth. Wait, when you multiply two exponents with the same base, don't you add the exponents? Let me check. Yes, the rule is a^m * a^n = a^(m+n). So here, 2 + 3 is 5, right? So that should be a^5, not a^6. So A is incorrect. Got it.Moving on to option B: a squared plus a cubed equals a to the sixth. Hmm, addition. The exponent rules for addition are different from multiplication. With addition, you can't combine the exponents unless there's a common factor. For example, if you have a^2 + a^3, you can factor out an a^2, getting a^2(1 + a). But you can't add the exponents here. So a^2 + a^3 doesn't simplify to a^6. That doesn't make sense. So B is definitely wrong.Option C: a^8 divided by a^4 equals a^2. The division rule for exponents is a^m / a^n = a^(m-n). So 8 minus 4 is 4, so it should be a^4. But the answer here is a^2. That's not right. So C is incorrect.Now option D: (a^3)^2 equals a^6. Let's think. When you raise a power to another power, you multiply the exponents. So (a^m)^n = a^(m*n). Here, 3 times 2 is 6, so that's a^6. That matches the answer given. So D is correct.Let me just double-check each one to make sure I didn't make a mistake. A: Multiplying a^2 and a^3, add exponents: 2+3=5. So a^5 vs a^6. Nope, A is wrong.B: Adding a^2 and a^3. Can't combine exponents with addition. So stays as a^2 + a^3. Definitely not a^6. B is wrong.C: Dividing a^8 by a^4. Subtract exponents: 8-4=4. Answer should be a^4, not a^2. So C is wrong.D: Raising a^3 to the 2nd power. Multiply exponents: 3*2=6. So a^6. Correct. D is the right answer.I don't see any mistakes in my reasoning. All other options misuse the exponent rules, while D applies the power rule correctly. So the answer must be D.</think>To determine which calculation is correct, we analyze each option using exponent rules:- Option A: (a^{2} cdot a^{3}). When multiplying like bases, add exponents: (a^{2+3} = a^{5}). Claimed result is (a^{6}). Incorrect.- Option B: (a^{2} + a^{3}). Exponents cannot be combined via addition. The expression remains (a^{2} + a^{3}). Claimed result is (a^{6}). Incorrect.- Option C: (a^{8} div a^{4}). When dividing like bases, subtract exponents: (a^{8-4} = a^{4}). Claimed result is (a^{2}). Incorrect.- Option D: ((a^{3})^{2}). When raising a power to another power, multiply exponents: (a^{3 cdot 2} = a^{6}). Claimed result is (a^{6}). Correct.Answer: boxed{D},<think>Okay, so I need to figure out which of these calculations is correct. There are four options: A, B, C, and D. Let me go through each one step by step. Starting with option A: a squared multiplied by a cubed equals a to the sixth power. Hmm, I remember that when you multiply exponents with the same base, you add the exponents. So, a² * a³ should be a^(2+3) which is a^5. But option A says a^6. That doesn't seem right. Maybe I made a mistake? Let me check again. No, the rule is definitely add the exponents when multiplying. So 2 + 3 is 5, so A is incorrect.Moving on to option B: a squared plus a cubed equals a^6. Wait a minute, addition? Exponents don't add when you're adding terms. If the bases are the same and you're multiplying, you add exponents, but addition is different. For example, 2² + 2³ is 4 + 8 = 12, which is not 2^6 (which is 64). So unless there's some specific value of a that makes a² + a³ = a^6, which is not generally true. So B is probably wrong too.Option C: a^8 divided by a^4 equals a^2. Let me recall the division rule for exponents. When you divide like bases, you subtract the exponents. So a^8 / a^4 should be a^(8-4) = a^4. But the answer here is a^2. That doesn't match. So C is incorrect. Maybe they divided 8 by 4 instead of subtracting? That might be the mistake. So yeah, C is wrong.Now option D: (a³) squared equals a^6. This is a power raised to another power. The rule here is to multiply the exponents. So (a³)^2 = a^(3*2) = a^6. That's exactly what option D says. Let me verify with numbers to be sure. If a is 2, then (2³)^2 = 8² = 64, and 2^6 is also 64. So it checks out. Therefore, D is correct.Wait, let me just make sure I didn't miss anything. Let me recap:A: Multiplying exponents, add the powers. 2 + 3 = 5, not 6. Wrong.B: Adding terms, no exponent rule applies here. a² + a³ can't be simplified to a single exponent unless factoring, which isn't done here. So B is wrong.C: Dividing exponents, subtract the powers. 8 - 4 = 4, answer is 2. Wrong.D: Power of a power, multiply exponents. 3 * 2 = 6. Correct.Yep, that seems right. So the correct answer must be D.Final Answerboxed{D}</think>To determine which calculation is correct, we analyze each option:- Option A: (a^2 cdot a^3 = a^6) When multiplying exponents with the same base, we add the exponents: (a^2 cdot a^3 = a^{2+3} = a^5). This does not equal (a^6), so A is incorrect.- Option B: (a^2 + a^3 = a^6) Addition of terms does not follow exponent rules. There is no general rule where (a^2 + a^3) simplifies to a single exponent term like (a^6). Thus, B is incorrect.- Option C: (a^8 div a^4 = a^2) When dividing exponents with the same base, we subtract the exponents: (a^8 div a^4 = a^{8-4} = a^4). This does not equal (a^2), so C is incorrect.- Option D: ((a^3)^2 = a^6) When raising a power to another power, we multiply the exponents: ((a^3)^2 = a^{3 cdot 2} = a^6). This is correct.Thus, the correct answer is (boxed{D}).]

🔑:To solve this problem, we evaluate each option step by step:Option A: When multiplying powers with the same base, we add the exponents. Therefore, for a^{2}cdot a^{3}, the calculation should be:[a^{2}cdot a^{3} = a^{2+3} = a^{5}]So, option A is incorrect because it claims a^{2}cdot a^{3}=a^{6}.Option B: When adding powers, unless they are like terms (which in this case they are not because the exponents are different), we cannot simply add the exponents. Therefore, a^{2}+a^{3} does not simplify to a^{6} in any standard algebraic operation. So, option B is incorrect.Option C: When dividing powers with the same base, we subtract the exponents. Therefore, for a^{8}div a^{4}, the calculation should be:[a^{8}div a^{4} = a^{8-4} = a^{4}]So, option C is incorrect because it claims a^{8}div a^{4}=a^{2}.Option D: When raising a power to another power, we multiply the exponents. Therefore, for (a^{3})^{2}, the calculation should be:[(a^{3})^{2} = a^{3cdot2} = a^{6}]So, option D is correct because it correctly states (a^{3})^{2}=a^{6}.Therefore, the correct answer is boxed{D}.

🧠:[<think>Okay, so I need to find the area of triangle D E G given that G is the centroid of triangle ABC, which has an area of 24. D and E are midpoints of sides AB and BC respectively. Hmm, let's start by recalling some properties of centroids and midpoints in triangles.First, the centroid G divides each median into a ratio of 2:1, with the longer segment being closer to the vertex. Also, since D and E are midpoints, DE is a midline of the triangle ABC. The midline theorem states that the segment connecting the midpoints of two sides is parallel to the third side and half its length. So DE should be parallel to AC and half its length.Given that the area of triangle ABC is 24, maybe I can find the area of triangle DEB or something related first? Wait, but DE is a midline, so the area of triangle BDE would be a quarter of the area of ABC? Because if DE is half the length of AC and the height is also halved, then the area would be (1/2 * base) * (1/2 * height) = 1/4 * original area. So maybe the area of BDE is 6? Let me check.But actually, triangle BDE: BD is half of AB, BE is half of BC? Wait, D is the midpoint of AB, so BD is half of AB. E is the midpoint of BC, so BE is half of BC. The angle at B is the same for both triangles ABC and BDE. So the area of BDE would be (BD * BE * sin(theta)) / 2, while the area of ABC is (AB * BC * sin(theta)) / 2. So the ratio is ( (AB/2) * (BC/2) ) / (AB * BC) ) = 1/4. So yes, the area of BDE is 24 / 4 = 6. So area of BDE is 6.But the question is about triangle DEG. Hmm. So maybe I need to figure out where point G is in relation to triangle BDE.Since G is the centroid, it's located at the intersection of the medians. Let me try to visualize triangle ABC with D as the midpoint of AB and E as the midpoint of BC. Then the medians would be from A to the midpoint of BC, from B to the midpoint of AC, and from C to the midpoint of AB. Wait, D is the midpoint of AB, so the median from C to D. E is the midpoint of BC, so the median from A to E.But G is the centroid where all three medians intersect. So if I can figure out the coordinates of G relative to triangle BDE, maybe I can compute the area.Alternatively, maybe using coordinate geometry would help here. Let me set up coordinates for triangle ABC. Let me place point B at the origin (0,0), point C at (2c, 0), so that E, the midpoint of BC, is at (c, 0). Point A can be at (a, b). Then D, the midpoint of AB, is at ((a/2), (b/2)). The centroid G is the average of the coordinates of A, B, C: ( (a + 0 + 2c)/3, (b + 0 + 0)/3 ) = ( (a + 2c)/3, b/3 ).Now DE is the segment connecting D ((a/2, b/2)) and E (c, 0). Let me find the coordinates of G: ((a + 2c)/3, b/3). So the three points of triangle DEG are D (a/2, b/2), E (c, 0), and G ((a + 2c)/3, b/3).To find the area of triangle DEG, I can use the shoelace formula. Let's denote the coordinates as follows:D: (x1, y1) = (a/2, b/2)E: (x2, y2) = (c, 0)G: (x3, y3) = ((a + 2c)/3, b/3)The area is 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|Plugging in the values:1/2 | (a/2)(0 - b/3) + c(b/3 - b/2) + ((a + 2c)/3)(b/2 - 0) |Let's compute each term step by step.First term: (a/2)(0 - b/3) = (a/2)( -b/3 ) = -ab/6Second term: c(b/3 - b/2) = c( (2b - 3b)/6 ) = c( -b/6 ) = -bc/6Third term: ((a + 2c)/3)(b/2 - 0) = ((a + 2c)/3)(b/2) = (a + 2c)b /6So combining all three terms:- ab/6 - bc/6 + (ab + 2bc)/6Let's combine the numerators:[ -ab - bc + ab + 2bc ] /6Simplify:(-ab + ab) + (-bc + 2bc) = 0 + bc = bcTherefore, the area is 1/2 | bc/6 | = bc/12Now, we need to relate this to the area of triangle ABC, which is 24.What's the area of triangle ABC in terms of coordinates?Points are B (0,0), C (2c, 0), A (a, b). The area is 1/2 * base * height. The base BC is 2c, and the height is the vertical distance from A to BC, which is b (since BC is along the x-axis). Therefore, the area is 1/2 * 2c * b = bc. So bc = 24.Therefore, the area of triangle DEG is bc /12 = 24 /12 = 2.So the answer should be 2, which is option B.Alternatively, maybe there's a way to do this without coordinates. Let's see.Since D and E are midpoints, DE is a midline, so DE is parallel to AC and half its length. The centroid divides the median into 2:1. If we consider the centroid G, it's located 2/3 along the median from the vertex to the midpoint. So if we can figure out how G relates to DE.Alternatively, since DE is a midline, the area of ADE is 1/4 of ABC's area? Wait, no. If DE is parallel to AC and half its length, then the area of CDE would be 1/4 of ABC. Wait, maybe not.Wait, if DE is a midline, then the area of triangle BDE is 1/4 of ABC as we saw earlier, which is 6. Then perhaps the area of DEG is a fraction of that.Alternatively, maybe using mass point geometry. Since G is the centroid, the ratios are known.Alternatively, use vectors. Let's assign vectors to points. Let me try that.Let’s take point B as the origin. Let’s denote vector BA as a and vector BC as c. Then the coordinates of points:- B: 0- A: a- C: cMidpoint D is (A + B)/2 = a/2Midpoint E is (B + C)/2 = c/2Centroid G is (A + B + C)/3 = (a + c) /3Wait, centroid is the average of the three vertices. So if B is the origin, then centroid G is (A + C)/3.Wait, in vector terms, G = (A + B + C)/3. Since B is origin, G = (A + C)/3.So points:D: a/2E: c/2G: ( a + c ) / 3We need to find the area of triangle D E G.The area can be found using the cross product formula. The area is 1/2 | (E - D) × (G - D) |.Compute vectors E - D and G - D.E - D = c/2 - a/2 = ( c - a ) / 2G - D = ( a + c ) /3 - a/2 = ( 2(a + c) - 3a ) /6 = ( 2c - a ) /6Then the cross product (E - D) × (G - D) is [ ( c - a ) / 2 ] × [ ( 2c - a ) /6 ] = (1/12)( c - a ) × ( 2c - a )Compute the cross product:( c - a ) × ( 2c - a ) = c × 2c - c × a - a × 2c + a × aBut cross product of any vector with itself is zero, so c × c = 0 and a × a = 0. So:= 0 - c × a - 2a × c + 0 = -c × a - 2a × cBut a × c = -( c × a ), so:= -c × a - 2*(-c × a ) = -c × a + 2c × a = c × aTherefore, the cross product (E - D) × (G - D) = (1/12)( c × a )Therefore, the area is 1/2 * | (1/12)( c × a ) | = (1/24)| c × a |.But the area of triangle ABC is 24, which is 1/2 | a × c |. Because vectors BA = a and BC = c, so the area is 1/2 | a × c | = 24. Therefore, | a × c | = 48. Hence, the area of triangle DEG is (1/24)*48 = 2.So again, the area is 2, which is option B. That confirms the earlier result using coordinate geometry.Alternatively, using area ratios. Since DE is a midline, DE divides the triangle into smaller sections. The centroid divides the medians in 2:1. So maybe the area of triangle DEG can be found by considering the proportions.But I think both the coordinate and vector methods have given the answer 2. So the correct answer is B) 2.Final Answerboxed{B}</think>Given that ( G ) is the centroid of ( triangle ABC ) with an area of 24, and ( D ), ( E ) are the midpoints of sides ( AB ) and ( BC ) respectively, we need to find the area of ( triangle DEG ).1. Midline Theorem: ( DE ) is a midline of ( triangle ABC ), so ( DE parallel AC ) and ( DE = frac{1}{2}AC ). The area of ( triangle BDE ) is ( frac{1}{4} ) of the area of ( triangle ABC ), which is ( 6 ).2. Coordinates Setup: - Place ( B ) at the origin ((0, 0)), ( C ) at ((2c, 0)), and ( A ) at ((a, b)). - Midpoints: ( D ) is (left( frac{a}{2}, frac{b}{2} right) ) and ( E ) is ((c, 0)). - Centroid ( G ) is (left( frac{a + 2c}{3}, frac{b}{3} right) ).3. Shoelace Formula: - Coordinates of ( D ), ( E ), and ( G ): - ( D left( frac{a}{2}, frac{b}{2} right) ) - ( E (c, 0) ) - ( G left( frac{a + 2c}{3}, frac{b}{3} right) ) - Area calculation using shoelace formula: [ text{Area} = frac{1}{2} left| frac{a}{2}(0 - frac{b}{3}) + c(frac{b}{3} - frac{b}{2}) + frac{a + 2c}{3}(frac{b}{2} - 0) right| ] Simplifying the terms, we find the area is ( frac{bc}{12} ).4. Relate to Area of ( triangle ABC ): - The area of ( triangle ABC ) is ( 24 ), so ( bc = 24 ). - Therefore, the area of ( triangle DEG ) is ( frac{24}{12} = 2 ).5. Vector Cross Product: - Using vector cross product confirms the area calculation, resulting in ( 2 ).Thus, the area of ( triangle DEG ) is (boxed{B}).,<think>Okay, so I have this problem here: Given triangle ABC with area 24, G is the centroid, and D, E are midpoints of sides AB and BC respectively. I need to find the area of triangle DEG. The options are 1, 2, 3, or 4. Hmm, let me think step by step.First, I remember that the centroid of a triangle divides each median into a ratio of 2:1, with the longer part being closer to the vertex. So, maybe that will come into play here. Also, D and E are midpoints, so DE might be a midline of the triangle? Wait, DE connects the midpoints of AB and BC. Let me visualize the triangle: ABC, D is midpoint of AB, E is midpoint of BC. So connecting D and E... Hmm, in a triangle, the midline is parallel to the third side and half its length. But here, DE connects midpoints of two sides. Let me check: AB is one side, BC is another. Wait, the midline between AB and BC would actually be connecting D (midpoint of AB) to E (midpoint of BC). Is that considered a midline? Or is the midline usually between two sides connected by a common vertex?Wait, the midline theorem states that the segment connecting the midpoints of two sides is parallel to the third side and half its length. So in triangle ABC, connecting midpoints of AB and BC would create a segment DE. But is DE parallel to AC? Let me confirm. If D is the midpoint of AB and E is the midpoint of BC, then DE should be parallel to AC and half its length. Wait, no, maybe not. Let me think again.Actually, the midline connecting midpoints of AB and AC would be parallel to BC. Similarly, the midline connecting midpoints of AB and BC... Hmm, maybe I need to use coordinate geometry here to be precise.Let me assign coordinates to the triangle to make it easier. Let's place point B at the origin (0,0). Let me assume point A is at (2a, 0) so that D, the midpoint of AB, is at (a, 0). Then, let me place point C at (0, 2c) so that E, the midpoint of BC, is at (0, c). Wait, but then BC is from (0,0) to (0, 2c), so midpoint is (0, c). But then DE connects (a, 0) to (0, c). Hmm, is this correct?But maybe a better coordinate system. Let me choose coordinates such that the calculations are easier. Let's let ABC be a triangle with coordinates:Let’s set point B at (0, 0), point C at (2c, 0), so that E, the midpoint of BC, is at (c, 0). Then, point A can be at some coordinates (d, e). Then, D, the midpoint of AB, would be at ((d/2), (e/2)). Wait, maybe that's more complicated. Alternatively, maybe use specific coordinates where the area is 24. Let's see.Alternatively, let's take a coordinate system where point B is at (0,0), point C at (4,0), point A at (0,6). Then the area of triangle ABC is (base * height)/2 = (4*6)/2 = 12. Hmm, but the problem states the area is 24. So maybe double those coordinates? Let's make point B at (0,0), C at (8,0), A at (0,12). Then area is (8*12)/2 = 48. No, that's too big. Wait, maybe point A at (0,6), C at (8,0). Then area is (8*6)/2 = 24. Yes, that works.So let's fix coordinates:- Let B be at (0,0).- Let C be at (8,0).- Let A be at (0,6).Then, the area of triangle ABC is (base * height)/2 = (8 * 6)/2 = 24. Perfect.Now, D is the midpoint of AB. AB is from (0,6) to (0,0). The midpoint D is at (0,3).E is the midpoint of BC. BC is from (0,0) to (8,0). The midpoint E is at (4,0).Now, G is the centroid of triangle ABC. The centroid is the average of the coordinates of the three vertices. So:G_x = (0 + 8 + 0)/3 = 8/3 ≈ 2.666...G_y = (6 + 0 + 0)/3 = 6/3 = 2.So G is at (8/3, 2).Now, we need to find the area of triangle DEG, where D is (0,3), E is (4,0), and G is (8/3, 2).To find the area of triangle DEG, we can use the shoelace formula. Let's list the coordinates:D: (0, 3)E: (4, 0)G: (8/3, 2)So applying the shoelace formula:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Plugging in the coordinates:x1 = 0, y1 = 3x2 = 4, y2 = 0x3 = 8/3, y3 = 2Compute each term:First term: x1(y2 - y3) = 0*(0 - 2) = 0Second term: x2(y3 - y1) = 4*(2 - 3) = 4*(-1) = -4Third term: x3(y1 - y2) = (8/3)*(3 - 0) = (8/3)*3 = 8Sum of terms: 0 + (-4) + 8 = 4Take absolute value and divide by 2: |4| / 2 = 2So the area of triangle DEG is 2. That's option B.Wait, but let me verify this again because sometimes with centroids and midpoints, there might be a ratio or a proportion that I can use without coordinate geometry. Maybe there's a smarter way.Alternatively, since D and E are midpoints, DE is a midline. Wait, in the coordinate system above, DE was from (0,3) to (4,0). Is that a midline? Wait, midline connects midpoints of two sides, which DE does. In triangle ABC, midpoints of AB and BC. So DE is a midline, but not parallel to AC in this case? Wait, in the coordinate system, AC is from (0,6) to (8,0), so slope is (0 - 6)/(8 - 0) = -6/8 = -3/4. DE is from (0,3) to (4,0), slope is (0 - 3)/(4 - 0) = -3/4. Oh, same slope! So DE is parallel to AC and half its length. Since AC has length sqrt((8)^2 + (-6)^2) = sqrt(64 + 36) = sqrt(100) = 10. DE length is sqrt((4)^2 + (-3)^2) = sqrt(16 + 9) = 5. So DE is half of AC. That aligns with the midline theorem. So DE is parallel to AC and half its length.Now, the centroid G is located at 1/3 of the median from each vertex. So in this case, maybe we can consider the position of G relative to DE.Alternatively, since DE is a midline, the area of ADE would be 1/4 of the area of ABC? Wait, if DE is a midline, then the area of ADE is 1/4 of ABC? Wait, maybe not. Let me think. If DE is parallel to AC and half its length, and the distance between DE and AC is half the height. Then the area of trapezoid DECA would be the average of the two bases times height. Wait, maybe I need a different approach.Alternatively, in the coordinate system, we found the area of DEG is 2. But let's see if there's a way without coordinates.Since G is the centroid, it divides the medians into 2:1. Let's consider the medians. For example, in triangle ABC, the median from A to BC. But D is the midpoint of AB, E is the midpoint of BC.Wait, maybe connect the centroid to these midpoints and analyze the areas.Alternatively, using area ratios. Let's see:The centroid divides the triangle into six smaller triangles of equal area. Each has area 24/6 = 4. But is that applicable here?Wait, when you connect the centroid to the midpoints, maybe different areas come into play.Alternatively, note that DE is a midline, so the area of BDE would be 1/4 of ABC? Wait, no. The area of the triangle formed by midlines is 1/4 of the original. Wait, if you connect all three midpoints, you get the medial triangle, which has area 1/4 of the original. But here, DE is just one midline. The area related to DE would depend on how it's situated.Wait, in the coordinate system, DE is from (0,3) to (4,0). So triangle DEG also includes point G at (8/3, 2). Let me confirm the shoelace calculation again:Coordinates:D (0,3)E (4,0)G (8/3, 2)Compute area:Arrange the points in order, D, E, G.Apply shoelace formula:Sum1 = (0*0 + 4*2 + (8/3)*3) = 0 + 8 + 8 = 16Sum2 = (3*4 + 0*(8/3) + 2*0) = 12 + 0 + 0 = 12Area = |Sum1 - Sum2| / 2 = |16 - 12| / 2 = 4/2 = 2Yes, same result. So the area is 2, which is option B.Alternatively, maybe use vectors. Let's see.Let me assign vectors to the points. Let’s take B as the origin.Vector B = (0,0)Vector A = (0,6)Vector C = (8,0)Then, midpoint D of AB: (0 + 0)/2 = 0, (6 + 0)/2 = 3 ⇒ D = (0,3)Midpoint E of BC: (8 + 0)/2 = 4, (0 + 0)/2 = 0 ⇒ E = (4,0)Centroid G: ( (0 + 8 + 0)/3, (6 + 0 + 0)/3 ) = (8/3, 2), same as before.Then vectors for triangle DEG:D: (0,3)E: (4,0)G: (8/3, 2)To find area, compute the determinant:Area = (1/2)| (x_D(y_E - y_G) + x_E(y_G - y_D) + x_G(y_D - y_E)) |Plugging in:= (1/2)| 0*(0 - 2) + 4*(2 - 3) + (8/3)*(3 - 0) |= (1/2)| 0 + 4*(-1) + (8/3)*3 |= (1/2)| 0 -4 + 8 |= (1/2)|4| = 2Same result. So regardless of the method, the area is 2.Alternatively, using ratios. Since G is the centroid, and DE is a midline. Maybe we can find the height from G to DE and multiply by half the length of DE?Length of DE: as before, 5 units (from coordinates, distance between (0,3) and (4,0) is 5). So half of DE is 2.5, but wait, no. Wait, area is (base * height)/2.Alternatively, compute the distance from G to DE and use DE as the base.First, find the equation of line DE. Points D(0,3) and E(4,0). The slope is (0 - 3)/(4 - 0) = -3/4. So equation is y - 3 = (-3/4)(x - 0), so y = (-3/4)x + 3.Distance from G(8/3, 2) to this line.The formula for distance from point (x0, y0) to line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).First, rewrite DE equation in standard form:y = (-3/4)x + 3 ⇒ (3/4)x + y - 3 = 0 ⇒ 3x + 4y - 12 = 0 (multiplying both sides by 4).So a = 3, b = 4, c = -12.Distance from G(8/3, 2):|3*(8/3) + 4*2 - 12| / sqrt(3² + 4²)= |8 + 8 - 12| / 5= |4| / 5 = 4/5.So the distance from G to DE is 4/5. Then, the area of triangle DEG is (base * height)/2 = (DE * distance)/2.DE is 5, so area = (5 * 4/5)/2 = (4)/2 = 2.Same answer. So using distance formula also gives area 2.Therefore, the correct answer should be B) 2.Final Answerboxed{B}</think>Given that ( G ) is the centroid of (triangle ABC) with an area of 24, and ( D ), ( E ) are the midpoints of sides ( AB ) and ( BC ) respectively, we need to find the area of (triangle DEG ).1. Coordinates Assignment: - Place ( B ) at ((0, 0)), ( C ) at ((8, 0)), and ( A ) at ((0, 6)) to form (triangle ABC) with area 24. - Midpoints ( D ) and ( E ): - ( D ) (midpoint of ( AB )): ((0, 3)) - ( E ) (midpoint of ( BC )): ((4, 0)) - Centroid ( G ): - Coordinates: (left( frac{0 + 8 + 0}{3}, frac{6 + 0 + 0}{3} right) = left( frac{8}{3}, 2 right))2. Shoelace Formula: - Coordinates of ( D ), ( E ), ( G ): - ( D(0, 3) ) - ( E(4, 0) ) - ( G left( frac{8}{3}, 2 right) ) - Applying the shoelace formula: [ text{Area} = frac{1}{2} left| 0(0 - 2) + 4(2 - 3) + frac{8}{3}(3 - 0) right| ] [ = frac{1}{2} left| 0 + 4(-1) + 8 right| = frac{1}{2} left| -4 + 8 right| = frac{1}{2} left| 4 right| = 2 ]3. Verification: - Using the distance formula and line equation for ( DE ): - Equation of ( DE ): ( 3x + 4y - 12 = 0 ) - Distance from ( G left( frac{8}{3}, 2 right) ) to ( DE ): [ frac{|3 cdot frac{8}{3} + 4 cdot 2 - 12|}{sqrt{3^2 + 4^2}} = frac{|8 + 8 - 12|}{5} = frac{4}{5} ] - Area calculation using base ( DE ) (length 5) and height ( frac{4}{5} ): [ text{Area} = frac{1}{2} times 5 times frac{4}{5} = 2 ]Thus, the area of (triangle DEG) is (boxed{B}).,<think>Okay, so I need to find the area of triangle D E G given that G is the centroid of triangle ABC, which has an area of 24. D and E are midpoints of sides AB and BC respectively. Hmm, let me visualize this. First, let me recall that the centroid divides each median into a ratio of 2:1, with the longer segment being closer to the vertex. Also, since D and E are midpoints, DE should be a midline of the triangle, right? So DE is parallel to AC and half its length. That might be useful.Since the area of triangle ABC is 24, maybe I can use some properties of centroids and midpoints to find the area of triangle DEG. Let me try to break it down step by step.First, let's consider triangle ABC. The centroid G divides each median into parts with a ratio 2:1. Let me try to figure out coordinates for the points to make it easier. Maybe assigning coordinates to the vertices could help. Let's place point B at the origin (0,0). Then, let me assign coordinates to points A and C such that the calculations are manageable.Let’s suppose coordinates: Let’s set point B at (0,0). Let’s let AB be along the x-axis, so point A is at (2a, 0) for some a, and point C is somewhere in the plane, say (2c, 2d). Then, the midpoint D of AB would be at (a, 0), and the midpoint E of BC would be at (c, d). The centroid G is the average of the coordinates of A, B, and C. So, centroid G would be at ((2a + 0 + 2c)/3, (0 + 0 + 2d)/3) = ((2a + 2c)/3, (2d)/3) = (2(a + c)/3, 2d/3).Now, we need to find the area of triangle DEG. Points D, E, G have coordinates (a, 0), (c, d), and (2(a + c)/3, 2d/3) respectively. To find the area of triangle DEG, I can use the shoelace formula. The formula for the area given three points (x1,y1), (x2,y2), (x3,y3) is |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|.Plugging in the coordinates:x1 = a, y1 = 0x2 = c, y2 = dx3 = 2(a + c)/3, y3 = 2d/3So, compute:Area = |(a*(d - 2d/3) + c*(2d/3 - 0) + 2(a + c)/3*(0 - d))/2|Let me compute each term step by step.First term: a*(d - 2d/3) = a*(d/3) = ad/3Second term: c*(2d/3 - 0) = c*(2d/3) = 2cd/3Third term: 2(a + c)/3*(0 - d) = 2(a + c)/3*(-d) = -2d(a + c)/3Adding them up:ad/3 + 2cd/3 - 2d(a + c)/3Combine the terms:ad/3 + 2cd/3 - 2ad/3 - 2cd/3 = (ad/3 - 2ad/3) + (2cd/3 - 2cd/3) = (-ad/3) + 0 = -ad/3Take absolute value and divide by 2:|(-ad/3)| / 2 = (ad/3)/2 = ad/6So the area of triangle DEG is ad/6.Now, what's the area of triangle ABC? Let's compute that with the coordinates we set.Points A(2a,0), B(0,0), C(2c, 2d)Area of ABC is |(2a*(0 - 2d) + 0*(2d - 0) + 2c*(0 - 0))/2| = |(2a*(-2d) + 0 + 0)/2| = |(-4ad)/2| = | -2ad | = 2adBut the problem states the area of ABC is 24, so 2ad = 24 => ad = 12.Then, the area of triangle DEG is ad/6 = 12/6 = 2.Therefore, the answer should be 2, which is option B.Wait, let me verify if this approach is correct. Maybe I made a mistake with coordinates.Alternatively, is there a way to do this without coordinates?Since D and E are midpoints, DE is the midline, so DE is parallel to AC and half its length. The centroid divides the median in ratio 2:1. Let me consider the position of G relative to DE.But maybe another approach is using area ratios. Let's see.First, since D and E are midpoints, DE is midline, so the area of triangle BDE would be 1/4 of the area of ABC? Wait, no. If DE is a midline, then the area of BDE is 1/4 of ABC. Wait, triangle BDE: BD is half AB, BE is half BC (since E is the midpoint of BC), and angle at B is the same. So area would be (1/2 * base * height) compared to ABC. Since both base and height are halved? Wait, no. Wait, if DE is the midline, then triangle BDE is similar to triangle BAC with scale factor 1/2. Wait, no. Wait, DE is parallel to AC and half its length. So triangle BDE has base DE and height relative to DE equal to half the height of AC. Wait, maybe.Alternatively, the area of triangle BDE is 1/4 of ABC's area. Let's check:If DE is midline, then the area of BDE is 1/4 of ABC. Let me see. If the area of ABC is 24, then area of BDE would be 6? Wait, not sure. Let me think.Alternatively, since D and E are midpoints, then BD = DA = AB/2, BE = EC = BC/2. Then, the area of triangle BDE would be (BD/AB)*(BE/BC)* area of ABC. Since BD/AB = 1/2, BE/BC = 1/2, so area ratio is 1/2 * 1/2 = 1/4. So area of BDE is 24 * 1/4 = 6. Then, area of BDE is 6.But how does G relate to this?G is centroid, so it's located at the intersection of the medians. Let's consider the medians. Let's say, in triangle ABC, the medians are from A to midpoint of BC, from B to midpoint of AC, and from C to midpoint of AB. But D is the midpoint of AB, E is the midpoint of BC.Wait, so DE is the midline, which connects midpoints of AB and BC, so DE is parallel to AC and half its length, as we said.Now, centroid G is located 2/3 along the median from the vertex. Let's try to see where G is relative to triangle BDE.Alternatively, maybe we can find coordinates again but with specific values. Let's choose coordinates such that calculations are easier.Let me place point B at (0,0), point C at (2,0), point A at (0, 4). Then, the area of triangle ABC is (base * height)/2 = (2 * 4)/2 = 4, but we need area 24. So maybe scale up. Let me take A at (0, 12), B at (0,0), C at (4, 0). Then area is (4 * 12)/2 = 24. Perfect.So coordinates:A(0,12), B(0,0), C(4,0)Midpoint D of AB: ( (0+0)/2, (12+0)/2 ) = (0,6)Midpoint E of BC: ( (0+4)/2, (0+0)/2 ) = (2,0)Centroid G: average of coordinates of A, B, C: ( (0 + 0 + 4)/3, (12 + 0 + 0)/3 ) = (4/3, 12/3) = (4/3, 4)So points D(0,6), E(2,0), G(4/3,4)Now, compute the area of triangle DEG using shoelace formula.Coordinates:D(0,6), E(2,0), G(4/3,4)Compute the area:Using formula |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Plug in:x1=0, y1=6x2=2, y2=0x3=4/3, y3=4Compute:0*(0 - 4) + 2*(4 - 6) + (4/3)*(6 - 0) = 0 + 2*(-2) + (4/3)*6 = 0 -4 + 8 = 4Take absolute value and divide by 2: |4|/2 = 2So area is 2. Which is option B. That matches my previous result.Alternatively, let's check with another coordinate system. Let me take another set to confirm.Suppose ABC is a right triangle with legs of length 6 and 8, area 24. Let’s set B at (0,0), A at (0,6), C at (8,0). Then area is (6*8)/2=24.Midpoint D of AB is (0,3), midpoint E of BC is (4,0). Centroid G is ((0 + 8 + 0)/3, (6 + 0 + 0)/3) = (8/3, 2)Points D(0,3), E(4,0), G(8/3,2)Compute area:Using shoelace formula:0*(0 - 2) + 4*(2 - 3) + (8/3)*(3 - 0) = 0 + 4*(-1) + (8/3)*3 = 0 -4 + 8 = 4Area is |4|/2 = 2. Again, same result. So the area is 2. So the answer is B.Alternatively, maybe there's a way without coordinates. Let's think.Since DE is a midline, it's parallel to AC and half its length. The centroid G divides the medians in 2:1. Let's consider the height from G to DE. Since DE is part of the midline, which is halfway up the triangle. The centroid is located 1/3 of the way from the base. Hmm, maybe.Alternatively, in triangle ABC, the centroid G is located at the intersection of the medians. Let's consider the median from B to the midpoint of AC. Let’s call the midpoint of AC as F. Then, the median BF. But D is the midpoint of AB, E is the midpoint of BC.Wait, perhaps using vectors or area ratios.Alternatively, note that DE is a midline, so the area of ADE is 1/4 of ABC, but no, DE is part of the midline. Wait, triangle ADE: since D is midpoint of AB, E is midpoint of BC. Hmm, not sure.Alternatively, in triangle ABC, connect midpoints D, E, F (midpoint of AC) to form the medial triangle DEF, which has area 1/4 of ABC. Wait, medial triangle has area 1/4 of the original. So area DEF is 24 / 4 = 6.But our triangle is DEG. G is centroid. How is G related to DEF?The centroid G is also the centroid of the medial triangle DEF. Wait, is that true? The centroid of the original triangle is the same as the centroid of the medial triangle?Wait, the centroid of the medial triangle DEF is the same as the centroid of ABC. Because the medial triangle is similar to ABC with a ratio of 1/2. The centroid of DEF would be the average of D, E, F, which are midpoints. So centroid G' of DEF is ((D_x + E_x + F_x)/3, (D_y + E_y + F_y)/3). But since D, E, F are midpoints, their coordinates are averages of the original vertices. So centroid G' would be (( (A_x + B_x)/2 + (B_x + C_x)/2 + (A_x + C_x)/2 ) / 3, similar for y). Calculating this would give (( (A_x + B_x + B_x + C_x + A_x + C_x ) / 2 ) / 3, ...) which simplifies to ((2A_x + 2B_x + 2C_x)/6, ...) = ((A_x + B_x + C_x)/3, ...), which is the centroid G of ABC. So centroid of DEF is same as centroid G of ABC.Therefore, in triangle DEF, centroid G is the same as centroid of ABC. So triangle DEF has area 6, and G is its centroid. Then, perhaps the area of DEG is a fraction of DEF.But triangle DEG is part of DEF. Let me see. In triangle DEF, points D, E, F are midpoints, and G is the centroid. So DEG is a triangle formed by two midpoints and centroid. How can I find its area?Alternatively, since DEF has area 6, and G is the centroid, maybe we can use barycentric coordinates or area ratios.Alternatively, in triangle DEF, G divides each median into 2:1 ratio. So, for example, if we take the median from D to midpoint of EF, which is the midpoint of EF. But maybe it's complicated.Alternatively, since DE is a side of DEF, and G is the centroid, which is located 2/3 along the median from each vertex. Wait, but in triangle DEF, the centroid G is 1/3 of the way from each side. Wait, but DEF is the medial triangle, so each side is half the length of ABC's sides, and the centroid is located at the same point as G in ABC.Alternatively, maybe area of DEG can be calculated as 1/3 of the area of DEF? Because in triangle DEF, connecting centroid would divide it into six small triangles of equal area. But maybe not. Wait, the centroid divides the triangle into three smaller triangles of equal area. So area of DGE would be 1/3 of DEF's area. Wait, DEF has area 6, so 6/3 = 2. Therefore, the area of DGE is 2. That gives the answer.So that's another way to see it. Since DEF is the medial triangle with area 6, and G is the centroid, which divides DEF into three smaller triangles each of area 2. Hence, area of DGE is 2. So answer is B.Therefore, through coordinate geometry and through area ratios, the answer is 2.Final Answerboxed{B}</think>Given that ( G ) is the centroid of (triangle ABC) with an area of 24, and ( D ), ( E ) are the midpoints of sides ( AB ) and ( BC ) respectively, we need to find the area of (triangle DEG).1. Coordinates Approach: - Assign coordinates to vertices ( A ), ( B ), and ( C ). For simplicity, place ( B ) at the origin ((0,0)), ( A ) at ((0, 12)), and ( C ) at ((4, 0)). This setup ensures the area of (triangle ABC) is 24. - Calculate midpoints ( D ) and ( E ): - ( D ) is the midpoint of ( AB ): ( D(0, 6) ) - ( E ) is the midpoint of ( BC ): ( E(2, 0) ) - Centroid ( G ) coordinates: ( Gleft(frac{0 + 0 + 4}{3}, frac{12 + 0 + 0}{3}right) = left(frac{4}{3}, 4right) ) - Use the shoelace formula to find the area of (triangle DEG): - Coordinates: ( D(0, 6) ), ( E(2, 0) ), ( Gleft(frac{4}{3}, 4right) ) - Area calculation: [ text{Area} = frac{1}{2} left| 0(0 - 4) + 2(4 - 6) + frac{4}{3}(6 - 0) right| = frac{1}{2} left| 0 - 4 + 8 right| = frac{1}{2} left| 4 right| = 2 ]2. Area Ratios Approach: - The medial triangle ( triangle DEF ) has an area of (frac{1}{4}) of (triangle ABC), which is ( 6 ). - The centroid ( G ) divides (triangle DEF) into three smaller triangles of equal area. - Therefore, the area of (triangle DEG) is (frac{1}{3}) of the area of (triangle DEF): [ text{Area of } triangle DEG = frac{6}{3} = 2 ]Both approaches confirm that the area of (triangle DEG) is (boxed{B}).,<think>Okay, so I have this geometry problem here. Let me read it again carefully. It says that G is the centroid of triangle ABC, which has an area of 24. D and E are the midpoints of sides AB and BC, respectively. Then I need to find the area of triangle DEG, with the options given being 1, 2, 3, or 4. Hmm. Alright, let me try to visualize this.First, let me recall that a centroid of a triangle divides each median into a ratio of 2:1, with the centroid being closer to the vertex than the midpoint. Also, since D and E are midpoints, DE is probably a midline of the triangle. Wait, the midline theorem states that the segment connecting the midpoints of two sides is parallel to the third side and half its length. So DE should be parallel to AC and half as long. That might be useful.Since the area of the whole triangle ABC is 24, maybe I can find the area of some smaller triangles and work my way towards triangle DEG. Let's start by sketching triangle ABC. Let me label the vertices A, B, C. Then D is the midpoint of AB, so AD = DB. Similarly, E is the midpoint of BC, so BE = EC. The centroid G is the intersection point of the medians. So the medians are from each vertex to the midpoint of the opposite side.So, the medians of triangle ABC would be from A to the midpoint of BC, which is E; from B to the midpoint of AC, let's call that point F; and from C to the midpoint of AB, which is D. The centroid G is where all three medians intersect.But wait, in the problem, they mention D and E as midpoints of AB and BC, so maybe I don't need the third midpoint F. Let me focus on the given points. DE is the midline connecting midpoints D and E, so as per the midline theorem, DE is parallel to AC and DE = (1/2)AC. That's correct.Now, centroid G divides each median into a ratio of 2:1. So if I take the median from B to the midpoint of AC (which is F), then BG:GF = 2:1. But maybe another median would be more helpful here. For instance, the median from A to E. Since E is the midpoint of BC, the median AE would be divided by centroid G into AG:GE = 2:1. Similarly, the median from C to D would be divided into CG:GD = 2:1.Hmm, maybe I need to figure out how the points D, E, and G are positioned relative to each other. Let me try to find coordinates for all these points. Coordinates might help me compute the area more straightforwardly.Let's assign coordinates to triangle ABC. Let me place point B at the origin (0,0) to simplify calculations. Then, let me let point C be at (2c, 0) since E is the midpoint of BC, which would then be at (c, 0). Similarly, let me place point A somewhere in the plane. Let's say A is at (a, b). Then, the midpoint D of AB would be at ((a/2), (b/2)).But maybe choosing specific coordinates would be better. Since the area is 24, maybe I can choose coordinates that make calculations easier. Let's try to set up the triangle so that calculations are straightforward.Let me assume triangle ABC is a coordinate system-friendly triangle. Let me place point B at (0,0), point C at (4,0), and point A at (0,6). Then, the area of triangle ABC would be (base * height)/2 = (4 * 6)/2 = 12, but wait, that's only 12, but we need an area of 24. Maybe I can double the sides. Let me try point C at (8,0) and point A at (0,6). Then the area would be (8 * 6)/2 = 24. Perfect. So let's assign coordinates:- B: (0,0)- C: (8,0)- A: (0,6)So, AB is from (0,6) to (0,0), BC is from (0,0) to (8,0), and AC is from (0,6) to (8,0). Let me confirm the area: the base BC is 8 units, the height is 6 units (from A to BC), so area is (8*6)/2 = 24. Correct.Now, let's find the midpoints D and E. D is the midpoint of AB. Since A is (0,6) and B is (0,0), midpoint D is ((0+0)/2, (6+0)/2) = (0,3). E is the midpoint of BC. B is (0,0) and C is (8,0), so midpoint E is ((0+8)/2, (0+0)/2) = (4,0).Now, we need to find centroid G. The centroid is the average of the coordinates of the three vertices. So, centroid G's coordinates are ((0 + 8 + 0)/3, (6 + 0 + 0)/3) = (8/3, 2). Wait, let's compute that again:x-coordinate: (A_x + B_x + C_x)/3 = (0 + 0 + 8)/3 = 8/3 ≈ 2.666...y-coordinate: (A_y + B_y + C_y)/3 = (6 + 0 + 0)/3 = 2.So, centroid G is at (8/3, 2). Alright.Now, we have points D (0,3), E (4,0), and G (8/3, 2). We need to find the area of triangle DEG.To find the area of triangle given three points, I can use the shoelace formula. The formula is:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Let me label the points:D: (0,3) = (x1, y1)E: (4,0) = (x2, y2)G: (8/3, 2) = (x3, y3)Plugging into the formula:Area = |(0*(0 - 2) + 4*(2 - 3) + (8/3)*(3 - 0))/2|Compute each term:First term: 0*(0 - 2) = 0*(-2) = 0Second term: 4*(2 - 3) = 4*(-1) = -4Third term: (8/3)*(3 - 0) = (8/3)*3 = 8So, sum of terms: 0 - 4 + 8 = 4Then, take absolute value (which is still 4) and divide by 2: 4/2 = 2So, the area is 2. Therefore, the answer should be option B) 2.Wait, but let me double-check. Maybe I made a mistake in coordinates or calculations. Let's verify the coordinates again.Points:A: (0,6), B: (0,0), C: (8,0). So, D is midpoint of AB: (0,3). E is midpoint of BC: (4,0). Centroid G: ((0+0+8)/3, (6+0+0)/3) = (8/3, 2). Correct.Coordinates for D, E, G are (0,3), (4,0), (8/3, 2). Applying shoelace formula:Arrange the points:D (0,3), E (4,0), G (8/3, 2)Compute using shoelace:List the coordinates in order, repeating the first at the end:0 34 08/3 20 3Compute sum of products down diagonals:(0*0 + 4*2 + (8/3)*3) - (3*4 + 0*(8/3) + 2*0)Compute each part:First part: 0 + 8 + 8 = 16Second part: 12 + 0 + 0 = 12Subtract: 16 - 12 = 4Take absolute value and divide by 2: |4|/2 = 2. So same result. Thus, area is 2. So answer is B) 2.Alternatively, maybe there's a way without coordinates. Let me think.Since DE is a midline, as we mentioned, DE is parallel to AC and half its length. So the length of DE is 1/2 AC, and the distance between DE and AC is half the height of the triangle.But how does centroid G relate to this?The centroid divides each median in 2:1 ratio. Let's consider the median from, say, A to E. Since E is the midpoint of BC, the median AE is divided by centroid G into AG:GE = 2:1.But perhaps we can use area ratios.The area of triangle ABC is 24. Since D and E are midpoints, the area of triangle BDE might be 1/4 of the original triangle? Wait, not sure. Wait, DE is a midline, so the area of triangle BDE? Wait, but DE is midline, so the area of the quadrilateral ADE would be?Wait, maybe breaking the triangle into smaller parts. Let me recall that connecting midpoints divides the triangle into four smaller triangles of equal area. But that's when connecting all three midpoints, forming the medial triangle. In this case, DE is one side of the medial triangle. The medial triangle has area 1/4 of the original. But triangle DEG is not the medial triangle, it's a smaller triangle within.Alternatively, since G is centroid, maybe we can express the coordinates in terms of vectors or use mass point geometry. But the coordinate method gave me 2, which is one of the options, so maybe that's correct. Let me see if there's another way to verify.Another approach: Since DE is a midline, the area of triangle ADE is 1/4 of the area of ABC. Wait, actually, the medial triangle (connecting all three midpoints) has area 1/4 of the original. But triangle ADE is half of the original triangle? Wait, maybe not. Let's see.If DE is parallel to AC and half its length, then the height from DE to AC is half the height of the original triangle. Therefore, the area of the trapezoid ACED would be (DE + AC)/2 * height_between_them. Wait, maybe this is complicating. Alternatively, the area of triangle BDE. Since DE is midline, triangle BDE is similar to triangle BAC with scale factor 1/2, so area would be (1/2)^2 = 1/4 of original area. But triangle BDE area would be 6? Since 24/4 = 6. But I'm not sure.Wait, if DE is midline, then the area of triangle BDE should be 1/4 of ABC's area. Let me check with coordinates. Points B (0,0), D (0,3), E (4,0). The area of triangle BDE can be calculated using shoelace:Coordinates: B(0,0), D(0,3), E(4,0)Area = |(0*(3 - 0) + 0*(0 - 0) + 4*(0 - 3))/2| = |0 + 0 + 4*(-3)/2| = |-12/2| = | -6 | = 6. So area is 6, which is indeed 24/4 = 6. So triangle BDE has area 6.But how does triangle DEG relate to this? Let's see. Points D (0,3), E (4,0), G (8/3, 2). We already calculated its area as 2. So from triangle BDE which is 6, to triangle DEG which is 2. How?Alternatively, maybe we can find the ratio of areas. Since G is the centroid, it's located 2/3 along the median from A to E. Wait, but which median? Let's see. From point A to E, which is the midpoint of BC. The centroid divides that median into AG:GE = 2:1. So if we consider the line from A (0,6) to E (4,0), the centroid G is 2/3 of the way from A to E. Similarly, the centroid is also located on other medians.But how does this relate to triangle DEG? Let's see. Maybe breaking down the problem into known areas. Let's consider the entire triangle divided into smaller parts by the medians and midlines.Since the centroid divides the triangle into six smaller triangles of equal area. Each of these smaller triangles has area 24/6 = 4. Wait, is that correct? Let me recall. When you draw all three medians, they divide the triangle into six smaller triangles, each of equal area. So each has area 4. Then, triangle DEG might be composed of parts of these smaller triangles.But I need to visualize which of these six triangles contains triangle DEG. Alternatively, since D and E are midpoints, maybe triangle DEG is within the medial triangle?Alternatively, let's use vectors. Let me assign vectors to the points. Let me take point B as the origin. Then, vector B is (0,0). Let me denote vectors:- Vector AB: from B(0,0) to A(0,6) is (0,6)- Vector BC: from B(0,0) to C(8,0) is (8,0) Then, midpoint D is (A + B)/2 = (0,3), midpoint E is (B + C)/2 = (4,0). Centroid G is (A + B + C)/3 = (0 + 0 + 8, 6 + 0 + 0)/3 = (8/3, 2), which matches earlier coordinates.To find the area of triangle DEG, we can use vectors. The area is 1/2 | cross product of vectors DE and DG |.First, find vectors DE and DG.Vector DE: E - D = (4 - 0, 0 - 3) = (4, -3)Vector DG: G - D = (8/3 - 0, 2 - 3) = (8/3, -1)Cross product in 2D is scalar magnitude given by (DE_x * DG_y - DE_y * DG_x)Compute that: (4)*(-1) - (-3)*(8/3) = (-4) - (-8) = (-4) + 8 = 4Then, area is 1/2 * |4| = 2. Same result. So that's consistent.Therefore, the area is indeed 2. So option B.Alternatively, another way: Since G is the centroid, it's located at 1/3 of the medians. So if I consider the coordinates again, G is (8/3, 2). So in the coordinate system we set up, moving from D (0,3) to E (4,0), and then to G (8/3, 2), forming triangle DEG.Alternatively, perhaps mass point geometry. But since we already have two methods giving area 2, and that's one of the answer choices, I think that's solid.But just to make sure, let's try to reason without coordinates. The key is that DE is a midline, so DE is parallel to AC and half its length. The centroid divides the median into 2:1. Let's say we take the median from A to E. The centroid G is 2/3 along that median from A. So from A to E, G is 2/3 from A. Since DE is a midline, which is halfway up the triangle. So the distance from DE to AC is half the height of the triangle. But how does G's position affect the height of triangle DEG?Alternatively, since DE is a midline, the area of ADE is 1/4 of ABC, so 6? Wait, in our coordinate system earlier, ADE would be triangle with points A (0,6), D (0,3), E (4,0). Let's compute its area.Using shoelace formula:A(0,6), D(0,3), E(4,0)Area = |(0*(3 - 0) + 0*(0 - 6) + 4*(6 - 3))/2| = |0 + 0 + 4*3/2| = |12/2| = 6. So area is 6, which is 24/4 = 6. So triangle ADE is 1/4 of the original triangle. Then, within triangle ADE, we have point G. The centroid of ABC. How is G positioned relative to triangle ADE?In our coordinate system, G is at (8/3, 2). So within triangle ADE, which has vertices at (0,6), (0,3), (4,0), the point G is inside that triangle. The area of triangle DEG is part of ADE. But how much?Alternatively, maybe using ratios. Since G divides the median AE into AG:GE = 2:1. So if I consider triangle ADE, which has area 6, then the line from G to D might divide ADE into smaller triangles. Let me see.But perhaps it's getting too vague. Since two methods with coordinates gave area 2, and that's an answer choice, I think confident enough that 2 is correct. So the answer is B) 2.Final Answerboxed{B}</think>Given that ( G ) is the centroid of ( triangle ABC ) with an area of 24, and ( D ), ( E ) are the midpoints of sides ( AB ) and ( BC ) respectively, we need to find the area of ( triangle DEG ).1. Coordinates Assignment: - Assign coordinates to the vertices of ( triangle ABC ): - ( B ) at ( (0, 0) ) - ( C ) at ( (8, 0) ) - ( A ) at ( (0, 6) ) - Calculate midpoints ( D ) and ( E ): - ( D ) is the midpoint of ( AB ): ( (0, 3) ) - ( E ) is the midpoint of ( BC ): ( (4, 0) ) - Calculate centroid ( G ): - ( G ) is the average of the coordinates of ( A ), ( B ), and ( C ): ( left( frac{0+0+8}{3}, frac{6+0+0}{3} right) = left( frac{8}{3}, 2 right) )2. Shoelace Formula: - Use the shoelace formula to find the area of ( triangle DEG ) with coordinates ( D(0, 3) ), ( E(4, 0) ), and ( G left( frac{8}{3}, 2 right) ): [ text{Area} = frac{1}{2} left| 0(0 - 2) + 4(2 - 3) + frac{8}{3}(3 - 0) right| ] [ = frac{1}{2} left| 0 + (-4) + 8 right| = frac{1}{2} left| 4 right| = 2 ]3. Verification: - The area calculation using vectors and cross product method confirms the result is consistent.Thus, the area of ( triangle DEG ) is (boxed{B}).]

🔑:2. B.S_{triangle D E G}=frac{1}{4} S_{triangle A C G}=frac{1}{12} S_{triangle A B C}=2 .

🧠:[<think>Okay, let's see here. I need to compute (2#0)#(1#4) where the operation a#b is defined as ab - a - 3. Hmm. Alright, first, maybe I should break this down step by step. So the problem has two parts: first, compute 2#0, then compute 1#4, and then use those results to compute the final value by combining them with the same # operation. Let me make sure I understand the order of operations here.So, according to the problem, the operator # takes two numbers, a and b, and returns ab - a - 3. The example they gave is 4#5 = 20 - 4 - 3 = 13. Let me check that: 4 times 5 is 20, minus 4 is 16, minus 3 is 13. Yep, that works. So the formula is indeed first multiply a and b, then subtract a, then subtract 3. Got it.Alright, so first compute 2#0. Let's substitute a=2 and b=0 into the formula. So that would be (2*0) - 2 - 3. Let me compute that. 2 times 0 is 0. Then subtract 2 gives -2, then subtract 3 gives -5. So 2#0 equals -5. Okay, that's straightforward.Next, compute 1#4. Using the same formula, a=1 and b=4. So (1*4) - 1 - 3. Let's calculate that. 1*4 is 4. Subtract 1 is 3, subtract 3 is 0. So 1#4 equals 0. Wait, that's interesting. So 1#4 is 0. Let me double-check that. 1 times 4 is 4. 4 minus 1 is 3, 3 minus 3 is 0. Yep, that's right.So now, we have to compute (-5)#0. Because (2#0) is -5 and (1#4) is 0. So substituting a=-5 and b=0 into the formula. Let's do that step by step. a#b = ab - a - 3. So (-5)*0 - (-5) - 3. Let's compute each part. (-5)*0 is 0. Then subtract (-5) is like adding 5. So 0 + 5 is 5. Then subtract 3 gives 2. Wait, so (-5)#0 is 0 - (-5) - 3. Which is 0 +5 -3 = 2. Correct. So (-5)#0 is 2. Therefore, the final result of (2#0)#(1#4) is 2.But hold on, let me just verify all the steps again to make sure I didn't make a mistake. It's easy to mess up signs, especially with negative numbers. So first step: 2#0. 2*0 =0, 0 -2= -2, -2 -3= -5. That's correct. Then 1#4. 1*4=4, 4-1=3, 3-3=0. That's right. Then combining those results: (-5)#0. Multiply -5 and 0, which is 0. Then subtract a, which is -5. So 0 - (-5) = +5. Then subtract 3, so 5-3=2. Yes, that's right. So the answer is 2.Alternatively, maybe there's another way to check. Let me redo the last operation again. (-5)#0 = (-5)(0) - (-5) -3. So that's 0 +5 -3 = 2. Yep, same result. I think that's solid.Is there a possibility that I misapplied the operator? Let's recall the definition: a#b = ab - a -3. So the first term is the product of a and b, then subtract a, then subtract 3. So in each case, when we do a#b, regardless of the order, it's always ab - a -3. So in the first case, 2#0: 2*0 -2 -3. Correct. Then 1#4: 1*4 -1 -3. Correct. Then combining (-5)#0: (-5)*0 - (-5) -3. Right. So no, the operator is applied correctly each time.Hmm, maybe the confusion would be if the operator is associative or not, but since we're explicitly told to compute (2#0)#(1#4), the order is specified. So first compute the left part, then the right part, then combine them. So there's no ambiguity in the order here.Alternatively, if someone tried to compute 2#0#1#4, but that's not the case here. The parentheses make it clear. So, yeah, (2#0) is one operand, and (1#4) is the other. Then those two results are combined with the # operator.Alright, so unless there's a miscalculation in my arithmetic, which I don't see, the answer should be 2. Let me verify once again.First operation: 2#0. 2*0=0. 0-2=-2. -2-3=-5. Correct.Second operation:1#4. 1*4=4. 4-1=3. 3-3=0. Correct.Third operation: (-5)#0. (-5)*0=0. 0 - (-5)=5. 5-3=2. Correct.Everything checks out. I think that's it. So the answer is 2.Final Answerboxed{2}</think>To compute ((2#0)#(1#4)) where (a # b = ab - a - 3):1. Compute (2 # 0): [ 2 # 0 = (2 cdot 0) - 2 - 3 = 0 - 2 - 3 = -5 ]2. Compute (1 # 4): [ 1 # 4 = (1 cdot 4) - 1 - 3 = 4 - 1 - 3 = 0 ]3. Use the results from steps 1 and 2 to compute ((-5) # 0): [ (-5) # 0 = (-5 cdot 0) - (-5) - 3 = 0 + 5 - 3 = 2 ]Thus, the final result is (boxed{2}).,<think>Okay, so I need to compute (2#0)#(1#4) where the operation a#b is defined as ab - a - 3. Let me take this step by step because I want to make sure I don't mix up the order or anything. First, I remember that when there are parentheses, I should solve the operations inside them first. So, I need to compute 2#0 and 1#4 separately and then combine those results using the same operation.Starting with 2#0. According to the definition, a#b is ab minus a minus 3. So, substituting a=2 and b=0, that would be (2*0) - 2 - 3. Let me compute that: 2*0 is 0, then subtract 2, which gives -2, and then subtract 3 more. So that's -2 - 3. Hmm, that's -5. Okay, so 2#0 equals -5. Got that part down.Now moving on to 1#4. Again, using the same formula. Here, a=1 and b=4. So it's (1*4) - 1 - 3. Calculating step by step: 1*4 is 4. Then subtract 1, which is 3, and subtract 3 more. So 3 - 3 is 0. Therefore, 1#4 equals 0. Alright, so now I have the two results: 2#0 is -5 and 1#4 is 0.Now the problem says to compute (2#0)#(1#4), which we now know translates to (-5)#0. Wait, hold on, let me verify. The first part was 2#0 = -5, and the second part was 1#4 = 0. So we need to perform the # operation on these two results: (-5)#0. Let me make sure I got that right. Yes, because the parentheses group the operations first, so after solving the inner parentheses, we have (-5)#0.Now, applying the definition again. Here, a is -5 and b is 0. So, a#b would be (-5)*0 - (-5) - 3. Let's compute each term. First, (-5)*0 is 0. Then subtract (-5), which is the same as adding 5. So now we have 0 + 5, which is 5. Then subtract 3. So 5 - 3 is 2. Therefore, (-5)#0 equals 2.Wait a second, that seems straightforward, but let me double-check to avoid any mistakes. So breaking it down again:a#b = ab - a - 3.So substituting a = -5 and b = 0:(-5)(0) - (-5) - 3.Calculating each term:First term: (-5)(0) = 0.Second term: - (-5) = +5.Third term: -3.So adding them up: 0 + 5 - 3 = 2. Yep, that checks out.Therefore, the result of (2#0)#(1#4) is 2.But let me just recap the entire process once more to ensure I haven't missed anything. Starting from the beginning:Problem: Compute (2#0)#(1#4) where a#b = ab - a - 3.Step 1: Compute 2#0.2#0 = (2)(0) - 2 - 3 = 0 - 2 - 3 = -5.Step 2: Compute 1#4.1#4 = (1)(4) - 1 - 3 = 4 - 1 - 3 = 0.Step 3: Now compute (-5)#0 using the results from steps 1 and 2.(-5)#0 = (-5)(0) - (-5) - 3 = 0 + 5 - 3 = 2.All steps lead to the conclusion that the answer is 2. I don't see any mistakes in the calculations. Each step follows the defined operation, and the arithmetic checks out. So I think that's the correct answer.Final Answerboxed{2}</think>To compute ((2#0)#(1#4)) where (a# b) is defined as (ab - a - 3):1. First, compute (2#0): [ 2#0 = (2 cdot 0) - 2 - 3 = 0 - 2 - 3 = -5 ]2. Next, compute (1#4): [ 1#4 = (1 cdot 4) - 1 - 3 = 4 - 1 - 3 = 0 ]3. Now, use the results from the previous steps to compute ((-5)#0): [ (-5)#0 = (-5 cdot 0) - (-5) - 3 = 0 + 5 - 3 = 2 ]Thus, the final result is (boxed{2}).]

🔑:1. First, compute (2 # 0): [ 2 # 0 = 2 cdot 0 - 2 - 3 = 0 - 2 - 3 = -5 ]2. Next, compute (1 # 4): [ 1 # 4 = 1 cdot 4 - 1 - 3 = 4 - 1 - 3 = 0 ]3. Now, use the results from steps 1 and 2 to compute ((-5) # 0): [ -5 # 0 = (-5) cdot 0 - (-5) - 3 = 0 + 5 - 3 = 2 ]The final answer is (boxed{2}).

🧠:[<think>Okay, let's tackle this problem step by step. So, the King wants to set a tax on chips to both reduce consumption and increase treasury revenue. We need to figure out if there's a tax rate that maximizes the revenue, and if so, what that maximum revenue would be.First, let's list out the given information:1. The market is perfectly competitive with linear demand and supply curves.2. A tax of 4.5 monetary units per unit results in tax revenue of 22.5.3. Doubling the tax to 9 monetary units increases tax revenue by 60%, so new revenue would be 22.5 * 1.6 = 36.4. Without any tax, consumption (quantity demanded) is 1.2 times higher than when the tax is 4.5. So, Q_untaxed = 1.2 * Q_taxed_4.5.5. Consumers only demand chips if the price is below 20 monetary units.Our goal is to determine if there's a tax rate that maximizes revenue, and if yes, find that maximum revenue.Let me start by recalling that in a perfectly competitive market with a per-unit tax, the tax incidence is shared between producers and consumers depending on the elasticities of demand and supply. The tax revenue is calculated as the tax per unit multiplied by the quantity sold after the tax.Given that demand and supply are linear, we can represent them as equations. Let's denote the demand curve as Qd = a - bP and the supply curve as Qs = c + dP. However, when a tax is imposed on producers, the supply curve shifts upward by the amount of the tax. So, the new supply curve becomes Qs = c + d(P - t), where t is the tax per unit.But actually, since it's a per-unit tax, the effective price the producers receive is P - t. So, the equilibrium occurs where Qd = Qs at the consumer price P, which includes the tax.Alternatively, we can model it as:Demand: Q = a - bPSupply (without tax): Q = c + dPWith a tax t, the supply becomes Q = c + d(P - t)Setting demand equal to supply with tax:a - bP = c + d(P - t)We can solve for equilibrium P and Q under tax t.Alternatively, we can use inverse demand and supply functions. Let me denote:Inverse Demand: P = α - βQInverse Supply (without tax): P = γ + δQWith a tax t, the inverse supply curve shifts up by t, so the new inverse supply becomes P = γ + δQ + tSetting inverse demand equal to inverse supply with tax:α - βQ = γ + δQ + tSolving for Q:α - γ - t = (β + δ)QSo,Q = (α - γ - t)/(β + δ)Then, the tax revenue is t * Q = t*(α - γ - t)/(β + δ)So, tax revenue is a quadratic function of t: R(t) = [t(α - γ - t)] / (β + δ)This is a quadratic equation in t, which will have a maximum at t = (α - γ)/2. So, maximum revenue occurs when t is half the difference between the demand intercept and supply intercept. Wait, that might be a key point here. If the tax revenue function is R(t) = t*Q(t), and Q(t) = (α - γ - t)/(β + δ), then R(t) is a quadratic in t: R(t) = (α - γ)t - t² all over (β + δ). So, it's a parabola opening downward, maximum at t = (α - γ)/2. But we need to check if this is feasible given the constraints, like the price must be less than 20.But maybe I need to find the actual demand and supply equations using the given data points. Let's try to do that.We have two tax rates and corresponding revenues:- At t1 = 4.5, revenue R1 = 22.5- At t2 = 9, revenue R2 = 36Also, when there's no tax (t0 = 0), the quantity Q0 = 1.2 * Q1, where Q1 is the quantity at t1 =4.5.Additionally, consumers only buy if price is less than 20, so the maximum price they are willing to pay is 20. So, the demand curve must hit P=20 at Q=0, or something like that. Wait, if the demand is positive only when price is less than 20, that suggests that the demand curve intersects the price axis at 20. So, the inverse demand curve is P = 20 - mQ, where m is the slope. Similarly, supply curve is something else.Wait, perhaps we can model the inverse demand as P = 20 - βQ, because at Q=0, P=20, and as Q increases, P decreases. Then the demand function is Qd = (20 - P)/β.Similarly, the supply curve could be P = γ + δQ, so Qs = (P - γ)/δ.But we need to determine these parameters β, γ, δ using the given information.Alternatively, let me denote:Let’s define the demand equation as Qd = a - bPSupply equation as Qs = c + dPBut when a tax t is imposed on suppliers, the supply equation becomes Qs = c + d(P - t). So, the equilibrium condition is:a - bP = c + d(P - t)Let me solve for P:a - bP = c + dP - dta - c + dt = (b + d)PP = (a - c + dt)/(b + d)Then, the equilibrium quantity Q is:Q = a - bP = a - b*(a - c + dt)/(b + d) = [a(b + d) - b(a - c + dt)]/(b + d)Simplify numerator:ab + ad - ab + bc - bdt = ad + bc - bdtSo,Q = (ad + bc - bdt)/(b + d) = [ad + bc - bdt]/(b + d)Alternatively, we can factor:Q = [ad + bc - bdt]/(b + d) = [ad + bc - t*bd]/(b + d) = [ad + bc]/(b + d) - [bd/(b + d)]*tSo, Q(t) = Q0 - [bd/(b + d)]*t, where Q0 is the quantity when t=0, which is (ad + bc)/(b + d)Wait, but we also know that when t=0, Q0 = 1.2 * Q1, where Q1 is the quantity when t=4.5.Given that, Q0 = 1.2 * Q(4.5)Similarly, we can write tax revenue as R(t) = t * Q(t)Given that at t=4.5, R=22.5, so 4.5 * Q(4.5) = 22.5 => Q(4.5) = 22.5 / 4.5 = 5Thus, Q(4.5) = 5Therefore, Q0 = 1.2 * 5 = 6So, when there's no tax, the quantity is 6.Also, when t=4.5, Q=5, and when t=9, Q(t=9) = ?But we also know that at t=9, revenue R=36, so 9 * Q(9) = 36 => Q(9) = 4So, Q(9) = 4Therefore, we have three points:- t=0: Q=6- t=4.5: Q=5- t=9: Q=4Since Q(t) is linear in t, as we saw earlier (Q(t) = Q0 - [bd/(b + d)]*t ), this is a linear relationship.From the three points, we can confirm that the slope is (5 - 6)/(4.5 - 0) = -1/4.5 ≈ -0.222...Similarly, between t=4.5 and t=9, Q goes from 5 to 4, so slope is (4 -5)/(9 -4.5) = -1/4.5, same as before. So, indeed Q(t) is linear in t with slope -1/4.5. So, Q(t) = 6 - (1/4.5)tHence, the equation for quantity as a function of tax is Q(t) = 6 - (2/9)t (since 1/4.5 is 2/9).So, Q(t) = 6 - (2/9)tThen, tax revenue R(t) = t * Q(t) = t*(6 - (2/9)t) = 6t - (2/9)t²This is a quadratic function, which opens downward, so it has a maximum at t = -b/(2a) for a quadratic equation ax² + bx + c. Here, a = -2/9, b = 6.So, vertex at t = -6/(2*(-2/9)) = -6 / (-4/9) = (6 * 9)/4 = 54/4 = 13.5So, the tax revenue is maximized at t =13.5 monetary units.But wait, the problem states that consumers have a positive demand only when price is less than 20. So, we need to ensure that the equilibrium price with tax t=13.5 is less than 20. Let's check that.First, let's find the equilibrium price as a function of tax t.From earlier, we have:Q(t) = 6 - (2/9)tBut also, from the inverse demand and supply functions.Alternatively, let's recall that in the absence of taxes, Q0 =6. Let's find the equilibrium price when t=0. That would be the price where Qd=Qs=6. Let's denote P0 as the equilibrium price without tax.But since we have Q(t) =6 - (2/9)t, which comes from the linear demand and supply. Also, we can find the relationship between price and tax.Alternatively, since Q(t) =6 - (2/9)t, and tax revenue is R(t) = t*Q(t), which we already maximized at t=13.5.But we need to check if at t=13.5, the price consumers pay is less than 20.Wait, how to find the price?From the demand function. If Q(t) =6 - (2/9)t, and since in a taxed equilibrium, the price that consumers pay is P(t), and the price producers receive is P(t) - t.Assuming that the demand curve is Qd = a - bP, so P = (a - Qd)/b. Let's find a and b.We know that when t=0, Q=6. So, in the absence of tax, the equilibrium quantity is 6. Let's denote the equilibrium price without tax as P0.From the supply side, the supply curve without tax is Qs = c + dP. At equilibrium, Qd=Qs=6. So,Demand: 6 = a - bP0Supply: 6 = c + dP0But we have another condition: when t=4.5, Q=5. Let's find the consumer price and producer price in that case.When t=4.5, the supply curve becomes Qs = c + d(P - 4.5). Setting equal to demand:a - bP = c + d(P -4.5)But we also know that when t=0:a - bP0 = c + dP0 =6So, from t=0:a - bP0 =6c + dP0 =6Thus, a - bP0 = c + dP0 => a - c = (b + d)P0But maybe we need more information. Alternatively, since we have the Q(t) =6 - (2/9)t, which is linear, and we can relate this to the demand and supply functions.Alternatively, the slope of Q(t) with respect to t is -2/9. Which is equal to -bd/(b + d) from earlier.Wait, earlier we had Q(t) = [ad + bc - bdt]/(b + d) = Q0 - [bd/(b + d)]*tBut we have Q(t) =6 - (2/9)t, so [bd/(b + d)] = 2/9So, bd/(b + d)=2/9But we need more equations to find b and d.Alternatively, since we know that the demand curve is such that at Q=0, the price is 20, as consumers have positive demand only when price is below 20. So, the demand curve intercept is P=20 when Q=0.Thus, the inverse demand equation is P =20 - mQSo, when Q=0, P=20. When Q=6 (no tax), P0=20 - m*6Similarly, the supply curve at t=0 gives Q=6 when P=P0. So, supply curve is P= n + kQ. At Q=6, P=P0. Also, when tax is applied, the supply curve shifts up by t.But we can relate the slope of the demand and supply curves. Let me try this approach.Let me denote:Inverse Demand: P =20 - mQInverse Supply (without tax): P =n + kQAt equilibrium without tax (t=0), Q=6, P=P0:20 - m*6 = n + k*6So, 20 -6m = n +6k ...(1)When tax t is applied, the new inverse supply is P =n + kQ + tSetting equal to inverse demand:20 - mQ =n + kQ + tSo, 20 -n - t = (m + k)QThus, Q = (20 -n - t)/(m +k)But from our previous result, Q(t) =6 - (2/9)tSo,(20 -n - t)/(m +k) =6 - (2/9)tMultiply both sides by (m +k):20 -n - t =6(m +k) - (2/9)t(m +k)Bring all terms to left:20 -n - t -6(m +k) + (2/9)t(m +k)=0This must hold for all t. Therefore, coefficients for t and constants must be zero.Coefficient of t:-1 + (2/9)(m +k) =0 => (2/9)(m +k)=1 => m +k= 9/2=4.5Constant term:20 -n -6(m +k)=0From m +k=4.5, substitute into constant term:20 -n -6*(4.5)=0 => 20 -n -27=0 => -n -7=0 => n= -7So, n= -7Now, from equation (1):20 -6m =n +6kWe have n= -7 and m +k=4.5 => k=4.5 -mSubstitute into equation (1):20 -6m = -7 +6*(4.5 -m)Calculate RHS: -7 +27 -6m=20 -6mSo,20 -6m=20 -6mWhich is an identity, so no new information. Thus, we need another equation to solve for m and k.But we already have m +k=4.5 and from inverse demand and supply.Wait, perhaps we can use the fact that when t=4.5, Q=5. But we already used that to get Q(t)=6 -2/9 t. Maybe we can find the prices at t=4.5 and relate to demand and supply.At t=4.5, Q=5. The price consumers pay is P_demand =20 -m*5The price suppliers receive is P_supply = P_demand -4.5=20 -m*5 -4.5=15.5 -5mBut also, from the supply curve (without tax), when Q=5, the price would have been P_supply =n +k*5= -7 +5kBut since with tax t=4.5, the suppliers receive P_supply=15.5 -5m, which should equal the original supply price at Q=5 minus tax? Wait, no.Wait, in the presence of tax t, the supply curve shifts up by t. So, the inverse supply with tax is P =n +kQ +tBut in equilibrium, P_demand = P_supply_with_tax =n +kQ +tBut Q=5 when t=4.5, so:P_demand =n +k*5 +4.5But also, P_demand=20 -m*5Thus,20 -5m= n +5k +4.5But n= -7, so:20 -5m= -7 +5k +4.5= -2.5 +5kSo,20 -5m +2.5=5k =>22.5 -5m=5k =>k=(22.5 -5m)/5=4.5 -mWhich is consistent with our earlier result that k=4.5 -m. So, again, we don't get new info.Thus, we need another relation. Maybe the equilibrium condition at t=0. At t=0, Q=6, P=20 -6m (from demand) and also P= n +6k= -7 +6k (from supply). So,20 -6m = -7 +6kBut since k=4.5 -m,20 -6m = -7 +6*(4.5 -m)Calculate RHS: -7 +27 -6m=20 -6mSo,20 -6m=20 -6mAgain, identity. So, we have infinite solutions for m and k as long as m +k=4.5. But maybe we don't need the exact values of m and k. Our main goal is to find the tax that maximizes revenue, which we already found as t=13.5. But we need to check if at t=13.5, the price consumers pay is still less than 20.But wait, at t=13.5, Q(t)=6 - (2/9)*13.5=6 - 3=3So, Q=3. Then, using the demand function, P=20 -m*3But we don't know m. However, since the maximum price is 20 when Q=0, and at Q=3, P=20 -3m. But we need to ensure that this price is less than 20, which it is as long as m>0, which it should be because demand slopes downward. So, the price would be 20 -3m. But we need to check if this is the price consumers pay, which it is. However, the problem states that residents have a positive demand only at a price lower than 20, which is already satisfied since at Q=3, P=20 -3m <20 (since m>0). So, that's okay.But wait, maybe the issue is whether the tax is so high that the equilibrium price exceeds 20. But since demand is zero when P>=20, if the tax causes the price to reach 20, then quantity demanded would be zero, and thus tax revenue would also be zero. So, we need to ensure that the tax t is such that the equilibrium price is less than 20.At the tax t_max that maximizes revenue, t=13.5, Q=3, so P=20 -3mBut we need to find m. Wait, but since we don't have enough info, perhaps m is determined by other means?Wait, from the supply side. At t=0, equilibrium price P0=20 -6m (from demand) and also P0= -7 +6k (from supply), and k=4.5 -m.So,P0=20 -6m= -7 +6*(4.5 -m)= -7 +27 -6m=20 -6mAgain, same equation. So, we can't determine m. It seems that m can be any value as long as k=4.5 -m. This suggests that there's a family of demand and supply curves that fit the given conditions, but they all result in the same Q(t) function, which is linear with Q(t)=6 - (2/9)t. Therefore, the tax revenue function R(t)=6t - (2/9)t² is the same regardless of m and k, as long as m +k=4.5.Therefore, even though we don't know the exact demand and supply slopes, the tax revenue function is determined, and its maximum is at t=13.5, yielding revenue R=6*13.5 - (2/9)*(13.5)^2Calculate this:First, 6*13.5=81Second, (2/9)*(13.5)^2= (2/9)*(182.25)= (2*182.25)/9= 364.5/9=40.5So, R=81 -40.5=40.5But we need to check if at t=13.5, the price is still below 20. But how?From the demand function: P=20 -mQAt t=13.5, Q=3, so P=20 -3mBut we don't know m. However, from the supply function, P= -7 +kQ +tWait, at t=13.5, the supply price (price received by producers) is P_s= -7 +kQ +t -t= -7 +kQWait, no. Wait, the inverse supply with tax is P= n +kQ +t= -7 +kQ +tBut in equilibrium, this equals the demand price: 20 -mQ= -7 +kQ +tAt t=13.5 and Q=3:20 -3m= -7 +3k +13.5Simplify:20 -3m=6.5 +3kBut k=4.5 -m, so:20 -3m=6.5 +3*(4.5 -m)=6.5 +13.5 -3m=20 -3mWhich simplifies to 20 -3m=20 -3m, an identity. So, no new information.Therefore, the price consumers pay at t=13.5 is P=20 -3m, but we can't determine m. However, since the demand must be positive only when P<20, and at t=13.5, Q=3>0, so P must be <20. Therefore, 20 -3m <20 => m>0, which is true since demand slopes downward. So, m>0, hence P=20 -3m <20. Therefore, it's valid.Therefore, the king can set a tax of 13.5 monetary units, and the maximum revenue collected would be 40.5 monetary units.But wait, in the given data, when t=4.5, R=22.5, and when t=9, R=36. So, increasing t from 4.5 to 9 increased revenue by 60%, which matches 22.5*1.6=36. Our model here gives R(t)=6t - (2/9)t². Let's check R(4.5):6*4.5=27; (2/9)*(4.5)^2=(2/9)*20.25=4.5; so R=27-4.5=22.5. Correct.R(9)=6*9=54; (2/9)*81=18; R=54-18=36. Correct.Thus, the model fits the given data. Then, at t=13.5, R=40.5. However, we need to confirm that at t=13.5, the price is still under 20.From the demand equation, since Q=3, P=20 -3m. Since m is positive, but we don't know its exact value. However, the problem states that residents have positive demand only when price is less than 20, but since Q=3 is positive, P must be less than 20. Therefore, regardless of m, as long as m>0, P=20 -3m <20. Thus, it's valid.Therefore, the king can set a tax of 13.5 monetary units, and the maximum revenue is 40.5.But let me check once more. Since we derived the revenue function based on the given data points and the linearity assumption, and since the maximum occurs at t=13.5, which gives a feasible price under 20, the answer should be yes, and the maximum revenue is 40.5.So, converting 40.5 into a boxed number: boxed{40.5}</think>The King of Consumerland can indeed set a tax that maximizes treasury revenue. Based on the given information, the tax revenue function is derived as ( R(t) = 6t - frac{2}{9}t^2 ), which is a quadratic function with a maximum at ( t = 13.5 ) monetary units. At this tax rate, the equilibrium quantity is 3 units, and the price remains below 20 monetary units, satisfying consumer demand constraints. The maximum tax revenue collected by the treasury is:[boxed{40.5}],<think>Okay, so the problem is about setting a tax on chips in Consumerland to both reduce consumption and increase treasury revenue. The king wants to know if he can set a tax that maximizes revenue and how much that would be. Let me try to break this down step by step.First, let's list out the given information:1. The market is perfectly competitive, so we're dealing with standard supply and demand curves.2. Demand and supply are linear and stable. That's helpful because linear equations are easier to work with.3. At a tax of 4.5 monetary units per unit, the tax revenue is 22.5.4. If the tax is doubled to 9 units, tax revenue increases by 60%, so that would be 22.5 * 1.6 = 36.5. Without any tax, consumption (quantity demanded) is 1.2 times higher than when the tax is 4.5. So, Q_untaxed = 1.2 * Q_taxed_at_4.5.6. Consumers only buy chips if the price is below 20 monetary units. That probably means the demand curve hits the price axis at 20.We need to find if there's a tax that maximizes revenue and what that maximum revenue is.Let me recall that tax revenue is calculated as the tax per unit (t) multiplied by the quantity sold after the tax (Q). So, Revenue = t * Q(t).In a perfectly competitive market with a per-unit tax on producers, the tax creates a wedge between the price consumers pay (P_d) and the price producers receive (P_s), where P_d = P_s + t.The equilibrium quantity with tax is where the quantity demanded at P_d equals the quantity supplied at P_s.Given that both supply and demand are linear, let's denote the demand curve as Q_d = a - b*P_d and the supply curve as Q_s = c + d*P_s. But since in equilibrium Q_d = Q_s = Q, we can write:Without tax: Q = a - b*P = c + d*PWith tax t: Q = a - b*(P_s + t) = c + d*P_sBut maybe it's better to use inverse demand and supply functions. Let's think again.Alternatively, let's denote the demand function as P = D(Q) and supply as P = S(Q). Without tax, equilibrium is D(Q) = S(Q). With a tax t on producers, the new supply curve becomes S(Q) + t, so equilibrium is D(Q) = S(Q) + t. The quantity with tax is Q_t, and the tax revenue is t * Q_t.But maybe to make it easier, let's use equations for demand and supply. Let's denote:Demand: P = D(Q) = m - n*QSupply: P = S(Q) = e + f*QBut with tax t, the effective supply curve becomes S(Q) + t = e + f*Q + t. So, setting D(Q) = S(Q) + t:m - n*Q = e + f*Q + tSolving for Q:m - e - t = (n + f) QSo, Q = (m - e - t)/(n + f)Tax revenue R(t) = t * Q = t*(m - e - t)/(n + f)So tax revenue is a quadratic function of t: R(t) = [ -t^2 + (m - e)t ] / (n + f)This is a quadratic that opens downward, so maximum revenue occurs at the vertex. The vertex is at t = (m - e)/2. So maximum revenue occurs at t = (m - e)/2, which is half the difference between the demand intercept and supply intercept. But we need to verify if that's within the feasible range of taxes where Q is positive and price is below 20.But given the problem, we need to find the parameters m, e, n, f. Let's see if the given data allows us to find these.Given that when t=4.5, R=22.5, so 4.5 * Q1 = 22.5 => Q1 = 5.When t=9, R=36, so 9*Q2 = 36 => Q2 = 4.Also, when t=0, Q0 = 1.2 * Q1 = 1.2*5 = 6.Additionally, consumers only buy if price is less than 20. So, the maximum price on the demand curve is 20. So, the demand curve intercept (when Q=0) is P=20. So m=20.Therefore, our demand function is P = 20 - n*Q.Supply function is P = e + f*Q.We have three equations from the given data:1. When t=4.5, Q=5. So, from equilibrium with tax:20 - n*5 = e + f*5 + 4.52. When t=9, Q=4. So:20 - n*4 = e + f*4 + 93. When t=0, Q=6. So:20 - n*6 = e + f*6So, three equations:1. 20 -5n = e +5f +4.5 → 20 -5n -5f -4.5 = e → 15.5 -5n -5f = e2. 20 -4n = e +4f +9 → 20 -4n -4f -9 = e → 11 -4n -4f = e3. 20 -6n = e +6f → e = 20 -6n -6fSo, set equations 1 and 2 equal to equation 3.From equation 1: 15.5 -5n -5f = 20 -6n -6fBring all terms to left:15.5 -5n -5f -20 +6n +6f = 0 → (-4.5) + n + f = 0 → n + f = 4.5From equation 2: 11 -4n -4f = 20 -6n -6fBring all terms to left:11 -4n -4f -20 +6n +6f =0 → (-9) +2n +2f =0 → 2n +2f =9 → n + f =4.5Which is the same as equation 1. So equations 1 and 2 give the same condition. Then equation 3 is e =20 -6n -6f. So we need another equation.Wait, we have two equations:From 1 and 2, we have n + f =4.5From equation 3: e =20 -6n -6f.But we need another equation. Wait, maybe we can use two of the three equations.Let me take equations 1 and 2 and set them equal via e.From equation 1: e=15.5 -5n -5fFrom equation 2: e=11 -4n -4fSet equal:15.5 -5n -5f =11 -4n -4f15.5 -11 = 5n +5f -4n -4f4.5 = n + fWhich is again the same. So the three equations only give us two unique equations: n + f =4.5 and e=20 -6n -6f.But we have three variables: e, n, f. So perhaps we need another relation.Wait, but maybe the intercept of the supply curve? Let me check.Wait, when Q=0, the supply price is e. But we don't have information about that. However, maybe when t is applied, the price consumers pay is P_d = S(Q) + t = e + f*Q + t. But the problem states that consumers only have positive demand for prices below 20. So the demand curve intercept is 20, which we already used.Alternatively, maybe we can proceed with the equations we have. Let's see.We have n + f =4.5Let me express f =4.5 -nThen, substitute into equation for e:e =20 -6n -6*(4.5 -n) =20 -6n -27 +6n=20 -27= -7So e= -7So the supply curve is P= -7 +f*QBut f=4.5 -nSo we still need another equation to find n. Wait, perhaps from the first equation.Wait, let's use the first equation with e=-7.From equation 1:20 -5n = e +5f +4.5We have e=-7, f=4.5 -nSo:20 -5n = -7 +5*(4.5 -n) +4.5Calculate right-hand side:-7 +22.5 -5n +4.5= (-7 +22.5 +4.5) -5n= 20 -5nSo left-hand side is 20 -5n = right-hand side 20 -5n. So identity, which doesn't help.Similarly, if we substitute into equation 2:20 -4n = e +4f +9With e=-7, f=4.5 -n:20 -4n = -7 +4*(4.5 -n) +9Right-hand side: -7 +18 -4n +9= ( -7 +18 +9 ) -4n= 20 -4nLeft-hand side 20 -4n = 20 -4n. Again, identity.Hmm, so it seems that with e=-7 and f=4.5 -n, we satisfy all equations. But we have infinitely many solutions because n can be any value, but since f must be positive (since supply curves usually slope upwards), f=4.5 -n >0 => n <4.5. Similarly, n must be positive because demand slopes downward, so n>0. So n is between 0 and4.5. But we need to find a specific value. Wait, perhaps there's another piece of information we haven't used yet.Wait, when there is no tax, Q=6. Then the equilibrium price is P= D(Q)=20 -n*6. Also, since equilibrium without tax is where D(Q)=S(Q), so 20 -n*6 = -7 +f*6. But since f=4.5 -n, substitute:20 -6n = -7 +6*(4.5 -n)Calculate right-hand side: -7 +27 -6n=20 -6nLeft-hand side is 20 -6n. So equal, which is again an identity. So this doesn't give us new info.Hmm. So maybe there is a missing piece? Wait, but all the given information is used. Wait, let's check:- We used the tax revenues at t=4.5 and t=9.- We used the fact that Q_untaxed=1.2*Q_taxed_at_4.5=1.2*5=6.- We used the price ceiling at 20.But maybe we need to ensure that when t is applied, the price consumers pay is less than 20. But since with tax t=4.5, Q=5. The price consumers pay is P_d =20 -n*5. Similarly, producers receive P_s = -7 +f*5. Let me compute these.Wait, with t=4.5, Q=5. Then P_d (price consumers pay) is 20 -n*5. The price producers receive is P_s = P_d - t =20 -n*5 -4.5 =15.5 -5n. But also, P_s = -7 +f*5. Since f=4.5 -n, this is -7 +5*(4.5 -n)= -7 +22.5 -5n=15.5 -5n. Which matches. So that's consistent.But since consumers have positive demand only if P <20, which is already satisfied here since P_d=20 -n*5. For that to be less than 20, 20 -5n <20 → -5n <0 → n>0, which is true as n is positive.But we don't have any further constraints. So perhaps the system is underdetermined? But that can't be. Wait, but the problem says that the king wants to know if he can set a tax to maximize revenue. Since tax revenue is R(t)= t*Q(t)= t*(20 - (-7) -t)/(n + f). Wait, from earlier, we had Q(t)= (m -e -t)/(n + f). Here, m=20, e=-7, so Q(t)= (20 - (-7) -t)/(n + f)= (27 - t)/(n + f). But n + f=4.5. So Q(t)= (27 - t)/4.5. Therefore, R(t)= t*(27 - t)/4.5.Wait, this is interesting. Because even though we couldn't find n and f individually, the sum n + f is 4.5, so Q(t)= (27 - t)/4.5.Therefore, tax revenue R(t) = t * Q(t) = t*(27 - t)/4.5 = (27t - t²)/4.5.So to maximize R(t), take derivative dR/dt=(27 -2t)/4.5. Set to zero:27 -2t=0 → t=13.5So maximum revenue occurs at t=13.5. But we need to check if this is feasible.But wait, when t=13.5, Q(t)= (27 -13.5)/4.5=13.5/4.5=3. So Q=3. But also, we need to check if the price consumers pay is less than 20.P_d =20 -n*Q(t). But Q(t)=3. However, we don't know n. Wait, but Q(t)= (27 - t)/4.5= (27 -13.5)/4.5=3. So P_d=20 -n*3. Since we need P_d <20, which is 20 -3n <20 → -3n <0 →n>0, which is true.But also, the price that producers receive is P_s= P_d - t= (20 -3n) -13.5=6.5 -3n. Also, P_s must be greater than or equal to the supply curve intercept, which is e=-7. Because supply is P= -7 +f*Q. When Q=0, P_s=-7. So as long as P_s >= -7, which it is because f is positive. But with Q=3, P_s= -7 +f*3. But P_s=6.5 -3n. Also, f=4.5 -n. So substituting:P_s= -7 + (4.5 -n)*3= -7 +13.5 -3n=6.5 -3n, which matches. So that's consistent.Therefore, even though we don't know n and f individually, the revenue function only depends on t, and we can compute the maximum at t=13.5. But wait, let's check with the given data.At t=4.5, Q=5. Then with t=13.5, Q=3. But when t=9, Q= (27 -9)/4.5=18/4.5=4, which matches the given data (since doubling tax to 9 leads to Q=4, which gives revenue 9*4=36, which is a 60% increase from 22.5). So all the given data is consistent with this model.Therefore, the tax that maximizes revenue is t=13.5. The maximum revenue is R=13.5*3=40.5.But wait, the problem states that consumers only have positive demand for prices below 20. Let's check what the price is when t=13.5.P_d=20 -n*3. But n +f=4.5, and f=4.5 -n. However, we need to check if P_d=20 -3n is less than 20. Which it is as long as n>0, which it is. So that's okay.But wait, let's check if the maximum tax revenue occurs at t=13.5. But in reality, tax revenue is R(t) = t*(27 - t)/4.5. So yes, this is a quadratic function with maximum at t=13.5. Therefore, the king can set a tax of 13.5 monetary units, and the maximum revenue will be 40.5.But let me verify with the given data points:At t=4.5: R=4.5*5=22.5 ✔️At t=9: R=9*4=36 ✔️ (which is 22.5*1.6=36)At t=0: Q=6, which is 1.2*5=6 ✔️So everything checks out.Therefore, the answer is yes, the king can set a tax that maximizes revenue, and the maximum revenue is 40.5 monetary units.But wait, in the problem statement, they mention "monetary units of Consumerland" and "monetary units per weight unit". So the answer should be in monetary units, which is 40.5. But let me check if the calculation is correct.R(t) = t*(27 -t)/4.5. At t=13.5:R=13.5*(27 -13.5)/4.5 =13.5*(13.5)/4.5=13.5*(3)=40.5. Correct.Therefore, the answer is 40.5.But the problem says "how much will be collected by the treasury?" So the answer is 40.5, which can be written as boxed{40.5}But let me double-check if there's a constraint I missed. For instance, when tax is 13.5, what's the price consumers pay?P_d=20 -n*3. But we don't know n. However, from the supply equation, when t=13.5, Q=3, and P_s= -7 +f*3. But f=4.5 -n.But also, P_s= P_d - t= (20 -3n) -13.5=6.5 -3n.From supply side: P_s= -7 + (4.5 -n)*3= -7 +13.5 -3n=6.5 -3n. So same as above. So regardless of n, these are consistent. Therefore, even though we don't know n, the calculation for revenue holds.Therefore, the answer is 40.5.Final Answerboxed{40.5}</think>Given the problem of setting a tax on chips in Consumerland to maximize treasury revenue, we start by analyzing the provided information:1. Tax Revenue at Different Rates: - At a tax rate of 4.5 monetary units, tax revenue is 22.5, implying the quantity sold is ( Q = frac{22.5}{4.5} = 5 ). - Doubling the tax to 9 units increases revenue by 60%, resulting in ( 22.5 times 1.6 = 36 ), implying the quantity sold is ( Q = frac{36}{9} = 4 ).2. Consumption Without Tax: - Without tax, consumption is 1.2 times higher than at a tax rate of 4.5, giving ( Q = 1.2 times 5 = 6 ).3. Linear Demand and Supply Functions: - The demand curve intercept is 20 monetary units (consumers only buy if price is below 20). - Using the equilibrium conditions with and without tax, we derive the linear equations for demand and supply.4. Key Equations: - Demand: ( P = 20 - nQ ) - Supply: ( P = -7 + fQ ) - Equilibrium with tax ( t ): ( Q = frac{27 - t}{4.5} ) (derived from the intercepts and slopes of demand and supply curves).5. Tax Revenue Function: - Revenue ( R(t) = t times Q(t) = t times frac{27 - t}{4.5} ). - This is a quadratic function ( R(t) = frac{27t - t^2}{4.5} ), which maximizes at ( t = frac{27}{2} = 13.5 ).6. Maximizing Revenue: - At ( t = 13.5 ), the quantity sold is ( Q = frac{27 - 13.5}{4.5} = 3 ). - Maximum revenue is ( R = 13.5 times 3 = 40.5 ).Verification with given data points confirms consistency. Therefore, the king can set a tax of 13.5 monetary units, resulting in maximum treasury revenue of:[boxed{40.5}]]

🔑:# Solution and Evaluation Criteria:Let the inverse demand function for chips be: P^{D}(Q)=a-b Q, and the inverse supply function for chips: P^{S}(Q)=c+d Q.Since with a per-unit tax of 4.5 monetary units, the tax revenue amounted to 22.5 monetary units, in equilibrium, 22.5 / 4.5 = 5 weight units of chips were consumed in this case. (maximum 1 point)Similarly, we can find how many chips were consumed with a tax of 2 * 4.5 = 9 monetary units:## 22.5 * 1.6 / 9 = 4 weight units. (maximum 1 point)Before the introduction of taxes, according to the condition, 1.2 * 5 = 6 weight units were consumed in equilibrium. (maximum 1 point)Considering the demand and supply functions, let's find how equilibrium on the market depends on the parameters of these functions before and after the introduction of the tax.Before the introduction of the tax: P^{D}(Q)=a-b Q^{*}=P^{S}(Q)=c+d Q^{*}, from which Q^{*}=(a-c) /(b+d) (maximum 1 point)After the introduction of the per-unit tax, in equilibrium, the difference between the price for buyers and the price sellers receive for their goods is the tax amount t, so: P^{D}(Q)=a-b Q^{t}-t=P^{S}(Q)=c+d Q^{t}, from which Q^{t}=(a-c-t) /(b+d) (maximum 1 point)Let's find the tax amount that will maximize tax revenues in the budget of Consumerland. For this, we need to find the maximum of the function: T=t Q^{t}=t(a-c-t) /(b+d), which represents a downward-opening parabola in (Q,T) coordinates. The maximum of such a function will be achieved at Q^{t}=(a-c) / 2(b+d) with a tax of t=(a-c) / 2. (maximum 4 points)It can be noted that with the introduction of a tax that maximizes its revenue, the equilibrium quantity of the good will be halved compared to what it was before the tax and will become equal to 6 / 2 = 3 weight units. (maximum 1 point)Considering the linearity of demand and supply (graphically, one can use the properties of similar triangles), the introduction of a tax proportionally reduces the equilibrium quantity of the good. That is, if with a tax of 9 monetary units, the equilibrium quantity decreased by 1.5 times compared to the equilibrium quantity without the tax, then to achieve an equilibrium quantity of 3 weight units, a tax of 9 * 1.5 = 13.5 monetary units must be introduced. (maximum 4 points)Then the revenue from such a tax, maximizing the budget revenues, will amount to 13.5 * 3 = 40.5 monetary units. (maximum 1 point)Each arithmetic error in the solution is penalized by a reduction of 1 point in the score.

🧠:[<think>Okay, let me try to figure out which of these three conclusions are correct. I need to check each one step by step. Let's start with the first conclusion.1. If proposition p is true, and proposition ¬q is true, then the proposition "p ∧ q" is true.Hmm. So, p is true, and ¬q is true. If ¬q is true, that means q is false, right? Because the negation of q being true would make q itself false. So, if p is true and q is false, then "p ∧ q" would be true only if both p and q are true. But since q is false here, "p ∧ q" should be false, not true. So the conclusion here says that "p ∧ q" is true, but actually it's false. Therefore, conclusion 1 is incorrect. Got that.Moving on to conclusion 2.2. The negation of the proposition "If xy = 0, then x = 0 or y = 0" is "If xy ≠ 0, then x ≠ 0 or y ≠ 0".Alright, let's recall how to negate an implication. The original statement is of the form "If A, then B", which is logically equivalent to "A → B". The negation of "A → B" is "A ∧ ¬B". So, applying that here, the original statement is "If xy = 0 (A), then x = 0 or y = 0 (B)". So, the negation should be "xy = 0 (A) ∧ ¬(x = 0 ∨ y = 0) (¬B)", which simplifies to "xy = 0 and x ≠ 0 and y ≠ 0". But the given negation here is "If xy ≠ 0, then x ≠ 0 or y ≠ 0", which is another implication. That doesn't seem right. The negation should not be another implication; it should be a conjunction. So conclusion 2 is incorrect. Wait, let me double-check. The original implication is "If A then B", negation is "A and not B". So in this case, the negation is "xy = 0 and it's not the case that x=0 or y=0", which is "xy=0 and x≠0 and y≠0". Therefore, conclusion 2 is wrong because it's phrased as another implication instead of a conjunction. So conclusion 2 is incorrect.Now, conclusion 3.3. The negation of "∀x ∈ ℝ, 2^x > 0" is "∃x ∈ ℝ, 2^x ≤ 0".Okay, this is about negating a universal quantifier. The rule is that the negation of "∀x, P(x)" is "∃x, ¬P(x)". So here, the original statement is "For all real numbers x, 2^x is greater than 0". The negation would be "There exists a real number x such that 2^x is not greater than 0", which is equivalent to "∃x ∈ ℝ, 2^x ≤ 0". That's exactly what conclusion 3 states. So conclusion 3 is correct.Wait, but let me verify with an example. 2^x is always positive for any real x, right? So the original statement is actually true, but the question isn't about the truth of the original statement, just about the form of its negation. Regardless of the actual truth value, the negation is correctly formed here. So yes, conclusion 3 is correct.So summarizing:1. Incorrect.2. Incorrect.3. Correct.Therefore, only one conclusion is correct. So the answer should be C: 1.But wait, let me check conclusion 2 again just to be thorough. The original statement is "If xy = 0, then x=0 or y=0". The user claims its negation is "If xy ≠ 0, then x ≠ 0 or y ≠ 0". Let's parse this.The original implication: A → B, where A is "xy = 0" and B is "x=0 ∨ y=0". The negation is A ∧ ¬B, which is "xy = 0 ∧ x ≠ 0 ∧ y ≠ 0". However, the given negation is "If xy ≠ 0, then x ≠ 0 or y ≠ 0", which is (¬A) → (¬B ∨ ¬C)? Wait, actually, the given negation is "If xy ≠ 0 (¬A), then x ≠ 0 (¬C) or y ≠ 0 (¬D)". Wait, the original B was "x=0 ∨ y=0", so ¬B would be "x≠0 ∧ y≠0". So the correct negation is "xy=0 and x≠0 and y≠0". But the user says the negation is "If xy ≠0, then x≠0 or y≠0", which is (¬A) → (¬C ∨ ¬D). However, in logic, "If ¬A, then ¬C ∨ ¬D" is equivalent to "¬A → (¬C ∨ ¬D)", which is equivalent to "A ∨ ¬C ∨ ¬D". This is not equivalent to the negation of the original implication. So conclusion 2 is definitely incorrect. So the user's negation is not correct. Hence conclusion 2 is wrong.Therefore, only conclusion 3 is correct. So the answer is C: 1.Final Answerboxed{C}</think>1. If proposition ( p ) is true, and proposition ( neg q ) is true, then the proposition ( p land q ) is true. - Given ( p ) is true and ( neg q ) is true (which means ( q ) is false), ( p land q ) should be false. Therefore, this conclusion is incorrect.2. The negation of the proposition "If ( xy = 0 ), then ( x = 0 ) or ( y = 0 )" is "If ( xy neq 0 ), then ( x neq 0 ) or ( y neq 0 )." - The original statement is ( xy = 0 rightarrow (x = 0 lor y = 0) ). The negation should be ( xy = 0 land neg (x = 0 lor y = 0) ), which simplifies to ( xy = 0 land x neq 0 land y neq 0 ). The given negation is another implication, which is incorrect. Therefore, this conclusion is incorrect.3. The negation of "( forall x in mathbb{R}, 2^x > 0 )" is "( exists x in mathbb{R}, 2^x leqslant 0 )." - The negation of a universal quantifier ( forall x, P(x) ) is ( exists x, neg P(x) ). The original statement is ( forall x in mathbb{R}, 2^x > 0 ), so the negation is correctly ( exists x in mathbb{R}, 2^x leqslant 0 ). Therefore, this conclusion is correct.The number of correct conclusions is 1. boxed{C},<think>Okay, let me try to figure out how many of these conclusions are correct. I'll take each one step by step.Starting with the first conclusion: "If proposition p is true, and proposition ¬q is true, then the proposition 'p ∧ q' is true." Hmm. Let's break this down. If p is true and ¬q is true, that means q is false, right? Because ¬q is the negation of q. So, if p is true and q is false, then p ∧ q would be true only if both p and q are true. But in this case, q is false, so p ∧ q should be false. Wait, the conclusion says that "p ∧ q" is true under these conditions. That doesn't seem right. For example, if p is "It is raining" and q is "It is sunny," then if it is raining (p true) and it is not sunny (¬q true, so q false), then "It is raining and it is sunny" would be false. So conclusion 1 is incorrect.Moving on to the second conclusion: "The negation of the proposition 'If xy = 0, then x = 0 or y = 0' is 'If xy ≠ 0, then x ≠ 0 or y ≠ 0'." Let me recall how to negate implications. The original statement is of the form "If A, then B", which is logically equivalent to "¬A ∨ B". The negation of that would be "A ∧ ¬B". So the negation should be "xy = 0 and not (x = 0 or y = 0)". Wait, the conclusion given here is another implication: "If xy ≠ 0, then x ≠ 0 or y ≠ 0". That doesn't seem right. Let me verify. The original implication is "A → B", where A is "xy = 0" and B is "x = 0 ∨ y = 0". The negation should be "A ∧ ¬B", so "xy = 0 and x ≠ 0 and y ≠ 0". But the given negation is "If xy ≠ 0, then x ≠ 0 or y ≠ 0", which is "¬A → (¬C ∨ ¬D)" where C is "x = 0" and D is "y = 0". That doesn't match the negation. So conclusion 2 is also incorrect.Now the third conclusion: "The negation of '∀x ∈ ℝ, 2^x > 0' is '∃x ∈ ℝ, 2^x ≤ 0'." Let's check this. The original statement is a universal quantification: "For all real numbers x, 2^x is greater than 0." The negation of a universal statement is an existential statement with the negation of the predicate. So ¬(∀x P(x)) is equivalent to ∃x ¬P(x). Here, P(x) is "2^x > 0", so ¬P(x) is "2^x ≤ 0". Therefore, the negation is indeed "There exists an x in ℝ such that 2^x ≤ 0". That seems correct. But wait, is 2^x ever less than or equal to 0 for real x? Let's see. 2^x is always positive for any real number x, right? So even though the negation is correctly formed, the statement "∃x ∈ ℝ, 2^x ≤ 0" is actually false. But the question is about the correctness of the negation, not the truth value of the negated statement. So regardless of whether the negated statement is true or false, the form of the negation is correct. Therefore, conclusion 3 is correct.So, out of the three conclusions, only the third one is correct. That means the number of correct conclusions is 1, which is option C.Final Answerboxed{C}</think>1. Analysis of Conclusion 1: - If ( p ) is true and ( neg q ) is true, then ( q ) is false. - The proposition ( p land q ) requires both ( p ) and ( q ) to be true. Since ( q ) is false, ( p land q ) is false. - Therefore, conclusion 1 is incorrect.2. Analysis of Conclusion 2: - The original proposition is "If ( xy = 0 ), then ( x = 0 ) or ( y = 0 )", which is logically equivalent to ( neg (xy = 0) lor (x = 0 lor y = 0) ). - The negation of an implication ( A rightarrow B ) is ( A land neg B ). Here, ( A ) is ( xy = 0 ) and ( B ) is ( x = 0 lor y = 0 ). - The correct negation is ( xy = 0 ) and ( x neq 0 ) and ( y neq 0 ). The given negation is another implication, which is incorrect. - Therefore, conclusion 2 is incorrect.3. Analysis of Conclusion 3: - The original statement is a universal quantification: ( forall x in mathbb{R}, 2^x > 0 ). - The negation of a universal statement is an existential statement with the negation of the predicate: ( exists x in mathbb{R}, 2^x leq 0 ). - The form of the negation is correct, regardless of the truth value of the negated statement. - Therefore, conclusion 3 is correct.The number of correct conclusions is 1. [boxed{C}]]

🔑:Analysis1. If "p land q" is true, then both p and q must be true. Thus, we need to evaluate based on this requirement.2. The negation of "or" is "and", and the negation of "for all" is "there exists". We solve this using the definition of negation.Solution1. If proposition p is true, and proposition neg q is true, it means q is false. Therefore, "p land q" is false. Hence, conclusion 1 is incorrect.2. The negation of the proposition "If xy = 0, then x = 0 or y = 0" is "If xy neq 0, then x neq 0 and y neq 0". Therefore, conclusion 2 is incorrect.3. The negation of "For all x in mathbb{R}, 2^x > 0" is "There exists x in mathbb{R}, 2^x leqslant 0". Therefore, conclusion 3 is correct.Thus, the correct answer is boxed{C}.