🧠:<think>Okay, so Ms. Rivera has this program to help first-generation immigrant students improve their math skills. She's comparing two groups: Group A, which gets her tailored support, and Group B, which follows the standard curriculum. She's tracking their test score improvements over the academic year. First, I need to figure out when Group A's improvement first exceeds Group B's by at least 10 points. The functions given are:- Group A: ( T_A(x) = 3x^2 + 2x + 5 )- Group B: ( T_B(x) = 2x^2 + 4x + 3 )Where ( x ) is the number of months since the program started. So, I think I need to find the smallest integer ( x ) where ( T_A(x) - T_B(x) geq 10 ).Let me subtract the two functions:( T_A(x) - T_B(x) = (3x^2 + 2x + 5) - (2x^2 + 4x + 3) )Simplify that:( 3x^2 - 2x^2 + 2x - 4x + 5 - 3 = x^2 - 2x + 2 )So, the difference in test score improvements is ( x^2 - 2x + 2 ). We need this to be at least 10:( x^2 - 2x + 2 geq 10 )Subtract 10 from both sides:( x^2 - 2x - 8 geq 0 )Now, solve the quadratic inequality. First, find the roots of the equation ( x^2 - 2x - 8 = 0 ).Using the quadratic formula:( x = frac{2 pm sqrt{(2)^2 - 4(1)(-8)}}{2(1)} = frac{2 pm sqrt{4 + 32}}{2} = frac{2 pm sqrt{36}}{2} = frac{2 pm 6}{2} )So, the roots are:( x = frac{2 + 6}{2} = 4 ) and ( x = frac{2 - 6}{2} = -2 )Since ( x ) represents months, it can't be negative, so we only consider ( x = 4 ).The quadratic ( x^2 - 2x - 8 ) opens upwards (since the coefficient of ( x^2 ) is positive), so it will be above zero when ( x leq -2 ) or ( x geq 4 ). Again, since ( x ) is positive, the inequality holds when ( x geq 4 ).Therefore, the test score improvement for Group A first exceeds that of Group B by at least 10 points in the 4th month.Wait, let me double-check. If I plug in ( x = 4 ):( T_A(4) = 3(16) + 2(4) + 5 = 48 + 8 + 5 = 61 )( T_B(4) = 2(16) + 4(4) + 3 = 32 + 16 + 3 = 51 )Difference: 61 - 51 = 10. Exactly 10 points. So, at 4 months, it's exactly 10 points. If the question is when it first exceeds by at least 10, then 4 is the correct answer.Now, moving on to the second part. The average initial scores are 50 for Group A and 55 for Group B. We need to find the month ( y ) when Group A's average surpasses Group B's.So, the total test scores for each group over time are:- Group A: Initial score 50 + improvement ( T_A(y) )- Group B: Initial score 55 + improvement ( T_B(y) )So, we need to find when:( 50 + T_A(y) > 55 + T_B(y) )Substitute the functions:( 50 + (3y^2 + 2y + 5) > 55 + (2y^2 + 4y + 3) )Simplify both sides:Left side: ( 50 + 3y^2 + 2y + 5 = 3y^2 + 2y + 55 )Right side: ( 55 + 2y^2 + 4y + 3 = 2y^2 + 4y + 58 )So, the inequality becomes:( 3y^2 + 2y + 55 > 2y^2 + 4y + 58 )Subtract ( 2y^2 + 4y + 58 ) from both sides:( y^2 - 2y - 3 > 0 )So, we have the quadratic inequality ( y^2 - 2y - 3 > 0 ). Let's find the roots:( y = frac{2 pm sqrt{4 + 12}}{2} = frac{2 pm sqrt{16}}{2} = frac{2 pm 4}{2} )Thus, roots are:( y = frac{2 + 4}{2} = 3 ) and ( y = frac{2 - 4}{2} = -1 )Again, since ( y ) is positive, we consider ( y = 3 ). The quadratic opens upwards, so the inequality ( y^2 - 2y - 3 > 0 ) holds when ( y < -1 ) or ( y > 3 ). Since ( y ) is positive, we look at ( y > 3 ).Therefore, Group A's average test score surpasses Group B's after 3 months. But let's check at ( y = 3 ):Group A: 50 + T_A(3) = 50 + (27 + 6 + 5) = 50 + 38 = 88Group B: 55 + T_B(3) = 55 + (18 + 12 + 3) = 55 + 33 = 88So, at 3 months, they are equal. The question is when Group A surpasses Group B, so it must be after 3 months. So, at 4 months:Group A: 50 + T_A(4) = 50 + 61 = 111Group B: 55 + T_B(4) = 55 + 51 = 106So, 111 > 106, so yes, at 4 months, Group A surpasses Group B.Wait, but the inequality suggested ( y > 3 ), so the smallest integer ( y ) is 4. So, the answer is 4 months.But hold on, in the first part, the improvement difference was 10 at 4 months, and here, the total scores cross at 3 months, but Group A surpasses Group B at 4 months. So, both answers are 4 months? That seems a bit coincidental.Wait, let me verify:For part 1, the improvement difference is 10 at 4 months.For part 2, the total scores cross at 3 months, but since at 3 months they are equal, the first time Group A surpasses is at 4 months.So, both answers are 4 months? Hmm.Alternatively, maybe I made a mistake in interpreting the first part. Let me see:In part 1, it's about the improvement, not the total score. So, the improvement for Group A is 10 points more than Group B at 4 months. In part 2, the total scores cross at 3 months, but Group A surpasses Group B at 4 months.So, both answers are 4 months? Or is there a different interpretation?Wait, no. In part 1, the improvement is 10 points more at 4 months. In part 2, the total scores cross at 3 months, but Group A surpasses Group B at 4 months. So, both answers are 4 months. That seems correct.But let me check the calculations again.For part 1:At x=4, T_A(4)=61, T_B(4)=51, difference=10.At x=3, T_A(3)=38, T_B(3)=33, difference=5. So, yes, 4 is the first month where the improvement is at least 10.For part 2:At y=3, total scores are 88 vs 88.At y=4, 111 vs 106. So, 4 is the first month where Group A surpasses Group B.So, both answers are 4 months. That seems correct.Final Answer1. The test score improvement for Group A first exceeds that of Group B by at least 10 points in month boxed{4}.2. The average test scores for Group A surpass those of Group B in month boxed{4}.</think>

🧠:<think>Okay, so I have these two probability problems to solve, and I need to figure them out step by step. Let me start with the first one.Problem 1: Probability of Finding First Edition BooksJack is visiting 10 bookshops, and in each shop, the probability of finding a first edition book is 0.3. He wants to find at least one first edition book in exactly 7 of these shops. I need to find the probability of this happening.Hmm, okay. So, this sounds like a binomial probability problem because each shop visit is an independent trial with two outcomes: success (finding at least one first edition) or failure (not finding any). The probability of success is 0.3, and we want exactly 7 successes out of 10 trials.Wait, but hold on. The problem says "at least one first edition book in exactly 7 of these shops." So, does that mean in exactly 7 shops, he finds at least one, and in the remaining 3, he finds none? Yes, that's what it sounds like.So, the probability of success in a single shop is 0.3, which is the probability of finding at least one first edition. Therefore, the probability of failure (not finding any) is 1 - 0.3 = 0.7.In binomial terms, the probability of exactly k successes in n trials is given by:P(k) = C(n, k) * p^k * (1-p)^(n-k)Where C(n, k) is the combination of n things taken k at a time.So, plugging in the numbers:n = 10, k = 7, p = 0.3.Therefore, P(7) = C(10, 7) * (0.3)^7 * (0.7)^3.Let me compute this step by step.First, compute C(10, 7). I know that C(n, k) = n! / (k! (n - k)! )So, C(10, 7) = 10! / (7! * 3!) = (10*9*8) / (3*2*1) = 120.Wait, let me verify that. 10! is 3628800, 7! is 5040, and 3! is 6. So, 3628800 / (5040 * 6) = 3628800 / 30240 = 120. Yes, that's correct.Next, compute (0.3)^7. Let me calculate that:0.3^1 = 0.30.3^2 = 0.090.3^3 = 0.0270.3^4 = 0.00810.3^5 = 0.002430.3^6 = 0.0007290.3^7 = 0.0002187So, (0.3)^7 ≈ 0.0002187.Then, (0.7)^3. Let me compute that:0.7^1 = 0.70.7^2 = 0.490.7^3 = 0.343So, (0.7)^3 = 0.343.Now, multiply all these together:P(7) = 120 * 0.0002187 * 0.343.First, multiply 120 and 0.0002187:120 * 0.0002187 = 0.026244.Then, multiply that by 0.343:0.026244 * 0.343 ≈ Let's compute this.0.026244 * 0.3 = 0.00787320.026244 * 0.04 = 0.001049760.026244 * 0.003 = 0.000078732Adding them together:0.0078732 + 0.00104976 = 0.008922960.00892296 + 0.000078732 ≈ 0.009001692So, approximately 0.009001692.Therefore, the probability is approximately 0.009, or 0.9%.Wait, that seems quite low. Let me double-check my calculations.First, C(10,7) is 120, correct.(0.3)^7 is 0.0002187, correct.(0.7)^3 is 0.343, correct.120 * 0.0002187 = 0.026244, correct.0.026244 * 0.343: Let me compute this more accurately.Compute 0.026244 * 0.343:First, 0.026244 * 0.3 = 0.00787320.026244 * 0.04 = 0.001049760.026244 * 0.003 = 0.000078732Adding these:0.0078732 + 0.00104976 = 0.008922960.00892296 + 0.000078732 = 0.009001692Yes, that's correct. So, approximately 0.009, or 0.9%.Hmm, that seems low, but considering that the probability of finding a first edition is only 0.3, getting 7 successes out of 10 is actually quite rare. So, maybe it's correct.Alternatively, maybe I should consider whether the problem is about finding at least one in exactly 7 shops, which is the same as exactly 7 successes. So, yes, that's correct.So, I think my calculation is correct. So, the probability is approximately 0.009, which is 0.9%.But let me express it more accurately. 0.009001692 is approximately 0.009002, so 0.9002%.So, to four decimal places, 0.0090.Alternatively, if I need to write it as a fraction, it's approximately 9/1000, but 0.009 is 9/1000.But maybe the question expects an exact fractional form.Wait, let me see:120 * (0.3)^7 * (0.7)^3.Expressed as fractions:0.3 is 3/10, so (3/10)^7 = 3^7 / 10^7 = 2187 / 10,000,000Similarly, 0.7 is 7/10, so (7/10)^3 = 343 / 1000So, P(7) = 120 * (2187 / 10,000,000) * (343 / 1000)Compute numerator: 120 * 2187 * 343Compute denominator: 10,000,000 * 1000 = 10,000,000,000First, compute numerator:120 * 2187 = Let's compute 120 * 2000 = 240,000, 120 * 187 = 22,440. So total is 240,000 + 22,440 = 262,440.Then, 262,440 * 343.Compute 262,440 * 300 = 78,732,000262,440 * 40 = 10,497,600262,440 * 3 = 787,320Adding these together:78,732,000 + 10,497,600 = 89,229,60089,229,600 + 787,320 = 90,016,920So, numerator is 90,016,920Denominator is 10,000,000,000So, P(7) = 90,016,920 / 10,000,000,000 = 0.009001692So, exactly, it's 0.009001692, which is approximately 0.009002, or 0.9002%.So, if I have to write it as a fraction, it's 90,016,920 / 10,000,000,000, which can be simplified.Divide numerator and denominator by 4: 22,504,230 / 2,500,000,000Divide by 2 again: 11,252,115 / 1,250,000,000Hmm, not sure if it reduces further. Maybe not necessary. So, the exact probability is 90,016,920 / 10,000,000,000, which is 0.009001692.So, approximately 0.009002.Therefore, the probability is approximately 0.9002%.I think that's the answer for the first problem.Problem 2: Expected Total Price of Three BooksNow, moving on to the second problem. Jack buys three first edition books, and their prices are modeled by random variables X, Y, Z with a joint probability density function f(x,y,z) = k(x² + y² + z²) over the region 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2. I need to find the normalization constant k and the expected total price Jack will pay.First, to find the normalization constant k, I need to ensure that the integral of f(x,y,z) over the entire region is equal to 1. That is,∫∫∫ f(x,y,z) dx dy dz = 1So,∫₀² ∫₀² ∫₀² k(x² + y² + z²) dx dy dz = 1Since the integrand is separable, I can split the integral into three separate integrals.First, factor out k:k * ∫₀² ∫₀² ∫₀² (x² + y² + z²) dx dy dz = 1Now, split the integral:k * [ ∫₀² x² dx ∫₀² dy ∫₀² dz + ∫₀² dx ∫₀² y² dy ∫₀² dz + ∫₀² dx ∫₀² dy ∫₀² z² dz ] = 1Wait, actually, no. Wait, the integral of (x² + y² + z²) over x, y, z is equal to the sum of the integrals of x², y², z² each multiplied by the integrals over the other variables.So, more precisely:∫₀² ∫₀² ∫₀² (x² + y² + z²) dx dy dz = ∫₀² x² dx ∫₀² dy ∫₀² dz + ∫₀² dx ∫₀² y² dy ∫₀² dz + ∫₀² dx ∫₀² dy ∫₀² z² dzEach of these terms is symmetric, so each will be equal.Compute one of them and multiply by 3.Compute ∫₀² x² dx ∫₀² dy ∫₀² dz.First, ∫₀² x² dx = [x³ / 3] from 0 to 2 = (8/3 - 0) = 8/3.Then, ∫₀² dy = 2, and ∫₀² dz = 2.So, the first term is (8/3) * 2 * 2 = (8/3)*4 = 32/3.Similarly, the second term is ∫₀² dx ∫₀² y² dy ∫₀² dz. That's the same as above, so also 32/3.Third term is ∫₀² dx ∫₀² dy ∫₀² z² dz, which is also 32/3.Therefore, the total integral is 3*(32/3) = 32.So, going back:k * 32 = 1 => k = 1/32.So, the normalization constant k is 1/32.Now, the second part is to find the expected total price Jack will pay for the three books.The total price is X + Y + Z, so the expected value E[X + Y + Z] = E[X] + E[Y] + E[Z].Since X, Y, Z are identically distributed (the joint density is symmetric in x, y, z), E[X] = E[Y] = E[Z]. So, E[X + Y + Z] = 3*E[X].Therefore, I just need to compute E[X] and multiply by 3.E[X] is the expected value of X, which is given by:E[X] = ∫₀² ∫₀² ∫₀² x * f(x,y,z) dx dy dzBut since f(x,y,z) = k(x² + y² + z²), and k = 1/32,E[X] = ∫₀² ∫₀² ∫₀² x * (1/32)(x² + y² + z²) dx dy dzAgain, we can split this integral:E[X] = (1/32) [ ∫₀² ∫₀² ∫₀² x³ dx dy dz + ∫₀² ∫₀² ∫₀² x y² dx dy dz + ∫₀² ∫₀² ∫₀² x z² dx dy dz ]Compute each term separately.First term: ∫₀² ∫₀² ∫₀² x³ dx dy dzCompute ∫₀² x³ dx = [x⁴ / 4] from 0 to 2 = 16/4 = 4.Then, ∫₀² dy = 2, ∫₀² dz = 2. So, first term is 4 * 2 * 2 = 16.Second term: ∫₀² ∫₀² ∫₀² x y² dx dy dzCompute ∫₀² x dx = [x² / 2] from 0 to 2 = 4/2 = 2.∫₀² y² dy = [y³ / 3] from 0 to 2 = 8/3.∫₀² dz = 2.So, second term is 2 * (8/3) * 2 = 32/3.Third term: ∫₀² ∫₀² ∫₀² x z² dx dy dzSimilarly, ∫₀² x dx = 2, ∫₀² z² dz = [z³ / 3] from 0 to 2 = 8/3, ∫₀² dy = 2.So, third term is 2 * 2 * (8/3) = 32/3.Therefore, putting it all together:E[X] = (1/32) [16 + 32/3 + 32/3]Compute the sum inside:16 is 48/3, so 48/3 + 32/3 + 32/3 = (48 + 32 + 32)/3 = 112/3.Therefore, E[X] = (1/32) * (112/3) = 112 / 96 = Simplify.Divide numerator and denominator by 16: 7 / 6.So, E[X] = 7/6.Therefore, E[X + Y + Z] = 3 * (7/6) = 7/2 = 3.5.But since the prices are in hundreds of pounds, the expected total price is 3.5 hundreds of pounds, which is 350 pounds.Wait, let me double-check my calculations.First, computing E[X]:E[X] = (1/32)[16 + 32/3 + 32/3] = (1/32)[16 + 64/3] = (1/32)[(48 + 64)/3] = (1/32)(112/3) = 112 / 96 = 7/6.Yes, that's correct.Therefore, the expected total price is 3*(7/6) = 7/2 = 3.5, which is 3.5 hundreds of pounds, so £350.Wait, but let me think again about the integrals.In the first term, ∫x³ dx dy dz: ∫x³ dx is 4, and ∫dy dz is 4, so 4*4=16.Second term: ∫x dx is 2, ∫y² dy is 8/3, ∫dz is 2, so 2*(8/3)*2=32/3.Third term is same as second term, so another 32/3.Total: 16 + 32/3 + 32/3 = 16 + 64/3 = (48 + 64)/3 = 112/3.Multiply by 1/32: 112/(3*32) = 112/96 = 7/6.Yes, correct.So, E[X] = 7/6, so E[X + Y + Z] = 3*(7/6) = 7/2 = 3.5.Therefore, the expected total price is £350.Wait, but the question says "the expected total price Jack will pay for the three books." Since X, Y, Z are in hundreds of pounds, so 3.5 hundreds of pounds is £350.Alternatively, if the question expects the answer in hundreds of pounds, it would be 3.5. But the problem says "find the expected total price Jack will pay for the three books," and since X, Y, Z are in hundreds of pounds, the total price is in hundreds of pounds, so 3.5.But let me check the wording:"Suppose the prices of the books are modeled as follows: Let X, Y, and Z be the random variables representing the prices of the three books in hundreds of pounds..."So, X, Y, Z are in hundreds of pounds. Therefore, the expected total price is E[X + Y + Z] in hundreds of pounds, which is 3.5. So, 3.5 hundreds of pounds is £350, but if they just want the expected value in the same units, it's 3.5.But the question says "find the expected total price Jack will pay for the three books." Since the units are in hundreds of pounds, the answer should be 3.5 hundreds of pounds, which is 350 pounds. But maybe the question expects the answer in hundreds, so 3.5.Wait, let me see: "the prices of the books are modeled as follows: Let X, Y, and Z be the random variables representing the prices of the three books in hundreds of pounds..."So, X, Y, Z are in hundreds of pounds. Therefore, the total price is X + Y + Z, which is also in hundreds of pounds. So, the expected value E[X + Y + Z] is in hundreds of pounds, which is 3.5. So, 3.5 hundreds of pounds is £350. But the question says "find the expected total price Jack will pay for the three books." It doesn't specify the units, but since X, Y, Z are in hundreds, the expected total price is 3.5 in hundreds, so 350 pounds.But maybe the answer is expected in hundreds, so 3.5. Let me check the problem statement again."Suppose the prices of the books are modeled as follows: Let X, Y, and Z be the random variables representing the prices of the three books in hundreds of pounds..."So, the prices are in hundreds, so the expected total price is in hundreds. So, 3.5 hundreds of pounds, which is 350 pounds. But in the answer, should I write 3.5 or 350?Wait, the question says "find the expected total price Jack will pay for the three books." Since the variables are in hundreds, the expected value is 3.5 hundreds, which is 350 pounds. So, perhaps the answer is 350 pounds, but written as 3.5 in hundreds.But in the problem statement, it's not clear whether the answer should be in hundreds or in pounds. Hmm.Wait, the joint density function is given for X, Y, Z in hundreds of pounds. So, the expected value E[X + Y + Z] is in hundreds of pounds. So, the expected total price is 3.5 hundreds of pounds, which is 350 pounds.But the question doesn't specify the units, so maybe it's safer to write both? Or perhaps just 3.5, since the variables are defined in hundreds.Wait, let me think. If X is in hundreds of pounds, then E[X] is in hundreds of pounds. So, E[X + Y + Z] is in hundreds of pounds. So, 3.5 hundreds of pounds is 350 pounds. But if the question asks for the expected total price, it's probably expecting the numerical value in pounds, so 350.But the problem says "find the expected total price Jack will pay for the three books." It doesn't specify units, but since X, Y, Z are in hundreds, the expected value is 3.5 in hundreds, so 350 pounds.Alternatively, maybe the answer is just 3.5, but in the context of hundreds, so 3.5 * 100 = 350.I think it's safer to write 350 pounds.But let me check the calculations again.E[X] = 7/6 ≈ 1.1667 hundreds of pounds.So, E[X + Y + Z] = 3*(7/6) = 7/2 = 3.5 hundreds of pounds, which is £350.Yes, that's correct.So, summarizing:Normalization constant k is 1/32.Expected total price is £350.Therefore, the answers are:1. Approximately 0.009002, or 0.9002%.2. Normalization constant k = 1/32, and expected total price is £350.But let me write the exact fraction for the probability in problem 1.Earlier, I had 90,016,920 / 10,000,000,000, which simplifies to 90,016,920 ÷ 10,000,000,000.Divide numerator and denominator by 4: 22,504,230 / 2,500,000,000Divide by 2: 11,252,115 / 1,250,000,000Not sure if it reduces further. Alternatively, as a decimal, it's 0.009001692.So, perhaps write it as 9001692/1000000000, but that's still a large fraction.Alternatively, leave it as 0.009002 or 0.9002%.But in the problem, it's better to write the exact value, which is 90,016,920 / 10,000,000,000, which can be simplified by dividing numerator and denominator by 4: 22,504,230 / 2,500,000,000, then divide by 2: 11,252,115 / 1,250,000,000.Not sure if it can be simplified further. Alternatively, write it as 0.009001692.But maybe the question expects the answer in terms of combinations and exponents, so perhaps leave it as 120*(0.3)^7*(0.7)^3, which is the exact expression.But I think the problem expects a numerical value, so 0.009002 or approximately 0.9002%.But to be precise, 0.009001692.So, rounding to four decimal places, 0.0090.Alternatively, as a fraction, 9001692/1000000000, but that's messy.Alternatively, 9001692 ÷ 1000000000 = 0.009001692.So, I think the answer is approximately 0.0090 or 0.90%.But let me check if I made a mistake in the combination.Wait, C(10,7) is 120, correct.(0.3)^7 is 0.0002187, correct.(0.7)^3 is 0.343, correct.Multiplying all together: 120 * 0.0002187 * 0.343 ≈ 0.009001692.Yes, correct.So, the exact probability is 0.009001692, which is approximately 0.9002%.Therefore, the answers are:1. The probability is approximately 0.009002, or 0.9002%.2. The normalization constant k is 1/32, and the expected total price is £350.But let me write the exact fraction for the probability.Wait, 0.009001692 is equal to 9001692/1000000000.Simplify:Divide numerator and denominator by 4: 2250423/250000000.Wait, 9001692 ÷ 4 = 2250423, and 1000000000 ÷ 4 = 250000000.2250423 and 250000000: Let's see if they have a common factor.2250423: Let's check divisibility by 3: 2+2+5+0+4+2+3 = 18, which is divisible by 3. So, 2250423 ÷ 3 = 750141.250000000 ÷ 3 is not an integer, so 3 is a factor of numerator but not denominator. So, the simplified fraction is 750141/83333333.333, which is not an integer. So, perhaps it's better to leave it as 9001692/1000000000 or 0.009001692.Alternatively, write it as 9001692/1000000000, which can be reduced by dividing numerator and denominator by 4, as above, but it's still messy.Therefore, I think it's better to present the probability as approximately 0.009002 or 0.9002%.So, to sum up:Problem 1: Probability ≈ 0.009002 or 0.9002%.Problem 2: Normalization constant k = 1/32, Expected total price = £350.Final Answer1. The probability is boxed{0.0090}.2. The normalization constant is boxed{dfrac{1}{32}} and the expected total price is boxed{350} pounds.</think>

🧠:<think>Alright, so I have this problem about a fine arts professor studying emotional responses in a gallery space. The function given is ( E(x, y, z) = e^{-(x^2 + y^2 + z^2)} ). I need to find the gradient vector and then use Lagrange multipliers to find critical points on a spherical surface. Hmm, okay, let's take it step by step.Starting with part 1: calculating the gradient vector (nabla E(x, y, z)). I remember that the gradient is a vector of the partial derivatives with respect to each variable. So, I need to find the partial derivatives of E with respect to x, y, and z.The function E is ( e^{-(x^2 + y^2 + z^2)} ). So, let's compute the partial derivative with respect to x first. The derivative of ( e^{u} ) with respect to x is ( e^{u} cdot du/dx ). Here, ( u = -(x^2 + y^2 + z^2) ), so ( du/dx = -2x ). Therefore, the partial derivative with respect to x is ( e^{-(x^2 + y^2 + z^2)} cdot (-2x) ), which simplifies to ( -2x e^{-(x^2 + y^2 + z^2)} ).Similarly, the partial derivatives with respect to y and z will be the same, just replacing x with y and z respectively. So, the partial derivative with respect to y is ( -2y e^{-(x^2 + y^2 + z^2)} ) and with respect to z is ( -2z e^{-(x^2 + y^2 + z^2)} ).Putting these together, the gradient vector (nabla E) is:[nabla E = left( -2x e^{-(x^2 + y^2 + z^2)}, -2y e^{-(x^2 + y^2 + z^2)}, -2z e^{-(x^2 + y^2 + z^2)} right)]Or factoring out the common term, we can write:[nabla E = -2 e^{-(x^2 + y^2 + z^2)} (x, y, z)]Now, interpreting this gradient. The gradient vector points in the direction of the steepest ascent of the function E. But since the coefficient here is negative, it actually points in the direction of steepest descent. So, at any point (x, y, z), the gradient tells us the direction in which the emotional response E decreases most rapidly. This makes sense because the function E is highest at the origin and decreases as we move away from the center. So, the gradient points towards the center, indicating where the emotional response is getting stronger.Moving on to part 2: finding the critical points of E on the spherical surface ( x^2 + y^2 + z^2 = R^2 ) using Lagrange multipliers. I remember that Lagrange multipliers are used to find local maxima and minima of a function subject to equality constraints.The method involves setting up the equation (nabla E = lambda nabla g), where g is the constraint function. Here, our constraint is ( g(x, y, z) = x^2 + y^2 + z^2 - R^2 = 0 ).First, let's compute the gradient of g. The partial derivatives of g with respect to x, y, and z are 2x, 2y, and 2z respectively. So,[nabla g = (2x, 2y, 2z)]From part 1, we have:[nabla E = -2 e^{-(x^2 + y^2 + z^2)} (x, y, z)]Setting up the Lagrange condition:[nabla E = lambda nabla g]So,[-2 e^{-(x^2 + y^2 + z^2)} (x, y, z) = lambda (2x, 2y, 2z)]Let me write this component-wise:1. ( -2 e^{-(x^2 + y^2 + z^2)} x = 2 lambda x )2. ( -2 e^{-(x^2 + y^2 + z^2)} y = 2 lambda y )3. ( -2 e^{-(x^2 + y^2 + z^2)} z = 2 lambda z )Hmm, okay. Let's see. Let's denote ( S = x^2 + y^2 + z^2 ), so ( e^{-S} ) is a common term. Also, note that the constraint is ( S = R^2 ), so ( e^{-S} = e^{-R^2} ), which is a constant.So, substituting ( e^{-S} = e^{-R^2} ), the equations become:1. ( -2 e^{-R^2} x = 2 lambda x )2. ( -2 e^{-R^2} y = 2 lambda y )3. ( -2 e^{-R^2} z = 2 lambda z )Let me simplify each equation by dividing both sides by 2:1. ( -e^{-R^2} x = lambda x )2. ( -e^{-R^2} y = lambda y )3. ( -e^{-R^2} z = lambda z )Now, let's rearrange each equation:1. ( (-e^{-R^2} - lambda) x = 0 )2. ( (-e^{-R^2} - lambda) y = 0 )3. ( (-e^{-R^2} - lambda) z = 0 )So, for each equation, either the coefficient is zero or the variable is zero.Case 1: ( -e^{-R^2} - lambda = 0 )Then, ( lambda = -e^{-R^2} )In this case, the equations are satisfied for any x, y, z, but we still have the constraint ( x^2 + y^2 + z^2 = R^2 ). So, any point on the sphere satisfies this condition if ( lambda = -e^{-R^2} ).But wait, if ( -e^{-R^2} - lambda = 0 ), then ( lambda = -e^{-R^2} ), so the other terms can be non-zero. But actually, if ( -e^{-R^2} - lambda = 0 ), then the equations reduce to 0 = 0, so x, y, z can be any values, but they must satisfy the constraint.Wait, that seems a bit confusing. Let me think again.Each equation is ( (-e^{-R^2} - lambda) x = 0 ). So, either ( (-e^{-R^2} - lambda) = 0 ) or x = 0. Similarly for y and z.So, two possibilities:1. ( (-e^{-R^2} - lambda) = 0 ), which implies ( lambda = -e^{-R^2} ). Then, the equations are satisfied for any x, y, z, but we still have the constraint ( x^2 + y^2 + z^2 = R^2 ). So, in this case, all points on the sphere are critical points? That doesn't seem right because E is a function that varies over the sphere.Wait, maybe not. Let me consider the other case.Case 2: If ( (-e^{-R^2} - lambda) neq 0 ), then x, y, z must be zero. But if x, y, z are zero, then the constraint ( x^2 + y^2 + z^2 = R^2 ) would require ( 0 = R^2 ), which is only possible if R = 0, but R is the radius of the sphere, so it's a positive constant. Therefore, this case only gives solutions if R = 0, which isn't the case here. So, the only possible case is Case 1, where ( lambda = -e^{-R^2} ).But then, does that mean every point on the sphere is a critical point? That can't be, because E is not constant on the sphere. Wait, E is ( e^{-R^2} ) on the sphere because ( x^2 + y^2 + z^2 = R^2 ), so E is constant on the sphere. Therefore, every point on the sphere is a critical point because the function is constant there.Wait, that makes sense. If E is constant on the sphere, then every point on the sphere is both a maximum and a minimum because the function doesn't change. So, all points on the sphere ( x^2 + y^2 + z^2 = R^2 ) are critical points, and the emotional response is constant at ( e^{-R^2} ) everywhere on the sphere.But let me double-check. If E is constant on the sphere, then yes, the gradient of E is zero on the sphere? Wait, no. Wait, the gradient of E is not zero on the sphere. The gradient is ( -2 e^{-S} (x, y, z) ), and on the sphere, S = R^2, so the gradient is ( -2 e^{-R^2} (x, y, z) ). But the gradient of E is not zero unless x, y, z are zero, which they aren't on the sphere.But in the Lagrange multiplier method, we set ( nabla E = lambda nabla g ). So, if ( nabla E ) is proportional to ( nabla g ), which is (2x, 2y, 2z). So, ( nabla E = -2 e^{-R^2} (x, y, z) ), and ( nabla g = 2 (x, y, z) ). So, ( nabla E = (-e^{-R^2}) nabla g ). Therefore, ( lambda = -e^{-R^2} ).So, this shows that for every point on the sphere, the gradient of E is proportional to the gradient of g, with proportionality constant ( lambda = -e^{-R^2} ). Therefore, every point on the sphere is a critical point.But since E is constant on the sphere, all these critical points are both maxima and minima. So, the function E doesn't have a unique maximum or minimum on the sphere; it's constant everywhere on the sphere.Wait, but E is ( e^{-R^2} ) on the sphere, which is a constant. So, yes, every point is both a maximum and a minimum because the function doesn't change. So, the emotional response is the same everywhere on the spherical surface.But let me think again. If E is constant on the sphere, then indeed, all points are critical points, and the function doesn't have a unique extremum. So, the maximum and minimum values of E on the sphere are both equal to ( e^{-R^2} ), occurring at every point on the sphere.So, summarizing, the critical points are all points on the sphere ( x^2 + y^2 + z^2 = R^2 ), and the emotional response is constant at ( e^{-R^2} ) everywhere on the sphere, meaning there's no variation in emotional response on the surface.But wait, that seems a bit counterintuitive because the gradient isn't zero, but the function is constant on the sphere. So, the gradient is tangent to the sphere? Wait, no. The gradient of E is pointing radially inward, as we saw in part 1. The gradient of g is also pointing radially outward. So, the gradient of E is a scalar multiple of the gradient of g, which makes sense because they are colinear vectors.So, in conclusion, all points on the sphere are critical points because the function E is constant on the sphere, so every point is both a maximum and a minimum. Therefore, the emotional response is uniform across the entire spherical surface.I think that's the answer. Let me just make sure I didn't make any mistakes in the calculations.Starting from the Lagrange condition:[nabla E = lambda nabla g]We found that ( lambda = -e^{-R^2} ), and this holds for all points on the sphere. Therefore, every point on the sphere satisfies the condition, making them all critical points. Since E is constant on the sphere, these are both maxima and minima.Yes, that seems correct.</think>

🧠:<think>Okay, so Raj is working on this project about polymer diffusion, and he has this partial differential equation to solve. The equation is the heat equation, which in this context describes how the concentration of a polymer diffuses over time. The equation is:[ frac{partial C}{partial t} = D nabla^2 C ]He’s given an initial concentration distribution:[ C(x, y, z, 0) = C_0 e^{-(x^2 + y^2 + z^2)} ]And he needs to find the concentration at any later time ( t ). Then, he also needs to figure out the rate of change of the total amount of polymer in a spherical region of radius ( R ) centered at the origin. Hmm, okay, let's break this down step by step.First, solving the PDE. Since the equation is linear and the initial condition is radially symmetric (it depends only on ( r = sqrt{x^2 + y^2 + z^2} )), it might be easier to work in spherical coordinates. But before jumping into that, maybe I can recall if there's a standard solution for the heat equation with a Gaussian initial condition.I remember that the heat equation in three dimensions with an initial Gaussian distribution has an analytic solution. The solution should also be Gaussian, but it spreads out over time. The general form is something like:[ C(r, t) = frac{C_0}{(1 + 4 D t / a^2)^{3/2}} e^{-r^2 / (1 + 4 D t / a^2)} ]Wait, but in the initial condition, the exponent is ( -(x^2 + y^2 + z^2) ), which is ( -r^2 ). So, comparing, it seems like ( a^2 = 1 ), so ( a = 1 ). Therefore, the solution simplifies to:[ C(r, t) = frac{C_0}{(1 + 4 D t)^{3/2}} e^{-r^2 / (1 + 4 D t)} ]Let me verify if this makes sense. At ( t = 0 ), it should reduce to the initial condition. Plugging ( t = 0 ), we get ( C(r, 0) = C_0 e^{-r^2} ), which matches. Good. As time increases, the denominator ( 1 + 4 D t ) increases, so the concentration spreads out, which makes sense for diffusion.Alternatively, another way to think about it is using the Fourier transform. The heat equation can be solved using Fourier transforms because it's a linear PDE with constant coefficients. The initial condition is a Gaussian, which is its own Fourier transform, so that might make things easier.Let me recall that the Fourier transform of ( e^{-a r^2} ) in three dimensions is another Gaussian. Specifically, in three dimensions, the Fourier transform of ( e^{-pi r^2} ) is itself, but scaling factors might differ. Let me check.The Fourier transform in 3D is:[ mathcal{F}{f(r)}(mathbf{k}) = int_{mathbb{R}^3} f(r) e^{-i mathbf{k} cdot mathbf{r}} d^3 r ]For ( f(r) = e^{-a r^2} ), the Fourier transform is:[ mathcal{F}{e^{-a r^2}}(mathbf{k}) = left( frac{pi}{a} right)^{3/2} e^{-pi^2 k^2 / a} ]Wait, maybe I need to adjust the constants. Let me compute it step by step.In 3D spherical coordinates, the Fourier transform becomes:[ mathcal{F}{e^{-a r^2}}(mathbf{k}) = int_0^infty e^{-a r^2} int_0^pi int_0^{2pi} e^{-i k r cos theta} sin theta dphi dtheta dr ]The angular integral can be separated. The integral over ( phi ) is ( 2pi ). The integral over ( theta ) is:[ int_0^pi e^{-i k r cos theta} sin theta dtheta = frac{2 sin(k r)}{k r} ]Wait, no, that's the integral of ( e^{i k r cos theta} ). Let me recall:The integral ( int_0^pi e^{i k r cos theta} sin theta dtheta ) is ( frac{sin(k r)}{k r} ) times 2? Wait, actually, it's ( frac{2 sin(k r)}{k r} ). But since we have ( e^{-i k r cos theta} ), it's the same because cosine is even.So, putting it all together, the Fourier transform becomes:[ 2pi times frac{2 sin(k r)}{k r} times int_0^infty e^{-a r^2} r dr ]Wait, no, hold on. Let me correct that. The integral over ( phi ) is ( 2pi ), the integral over ( theta ) is ( frac{2 sin(k r)}{k r} ), and then we have the radial integral:[ int_0^infty e^{-a r^2} r dr ]Wait, no, actually, the integral is:[ int_0^infty e^{-a r^2} left( int_0^pi e^{-i k r cos theta} sin theta dtheta right) r dr ]But the angular integral is ( frac{sin(k r)}{k r} times 2 ), so:[ 2 int_0^infty e^{-a r^2} frac{sin(k r)}{k r} r dr = frac{2}{k} int_0^infty e^{-a r^2} sin(k r) dr ]Hmm, now we have:[ frac{2}{k} int_0^infty e^{-a r^2} sin(k r) dr ]This integral can be evaluated using integration techniques. Let me recall that:[ int_0^infty e^{-a r^2} sin(k r) dr = frac{sqrt{pi}}{2 sqrt{a}} e^{-k^2 / (4a)} ]Yes, that's a standard integral. So substituting back:[ frac{2}{k} times frac{sqrt{pi}}{2 sqrt{a}} e^{-k^2 / (4a)} = frac{sqrt{pi}}{k sqrt{a}} e^{-k^2 / (4a)} ]Therefore, the Fourier transform of ( e^{-a r^2} ) is:[ mathcal{F}{e^{-a r^2}}(mathbf{k}) = frac{sqrt{pi}}{k sqrt{a}} e^{-k^2 / (4a)} ]Wait, but this seems a bit off because in 3D, the Fourier transform should have a dependence on ( k ) but not in the denominator like that. Maybe I made a mistake in the calculation.Alternatively, perhaps it's better to recall that in 3D, the Fourier transform of a radially symmetric function ( f(r) ) is also radially symmetric, and can be expressed in terms of the Hankel transform. The Hankel transform of order ( nu ) is used for functions defined on ( mathbb{R}^n ) with spherical symmetry.For the 3D Fourier transform, the Hankel transform of order ( nu = 1 ) is used. The formula is:[ mathcal{F}{f(r)}(k) = 2pi int_0^infty f(r) j_1(k r) r dr ]Where ( j_1 ) is the spherical Bessel function of the first kind of order 1. Hmm, this might complicate things, but perhaps it's manageable.Alternatively, maybe it's better to use the fact that the solution to the heat equation with a Gaussian initial condition is another Gaussian. Let me think about scaling.The heat equation in 3D is:[ frac{partial C}{partial t} = D nabla^2 C ]The general solution can be written using the Green's function, which for the heat equation is the fundamental solution. In 3D, the Green's function is:[ G(mathbf{r}, t) = frac{1}{(4 pi D t)^{3/2}} e^{-r^2 / (4 D t)} ]So, the solution is the convolution of the initial condition with the Green's function. Since the initial condition is also Gaussian, the convolution of two Gaussians is another Gaussian.Let me write the initial condition as:[ C(mathbf{r}, 0) = C_0 e^{-r^2} ]And the Green's function is:[ G(mathbf{r}, t) = frac{1}{(4 pi D t)^{3/2}} e^{-r^2 / (4 D t)} ]So, the solution at time ( t ) is:[ C(mathbf{r}, t) = int_{mathbb{R}^3} G(mathbf{r} - mathbf{r}', t) C(mathbf{r}', 0) d^3 r' ]Since both ( G ) and ( C ) are radially symmetric, this reduces to a radial convolution:[ C(r, t) = int_0^infty G(r - r') C(r') r'^2 dr' ]Wait, actually, in spherical coordinates, the convolution becomes:[ C(r, t) = int_0^infty G(r - r') C(r') r'^2 dr' ]But because both ( G ) and ( C ) are radially symmetric, we can use the fact that the convolution of two Gaussians is another Gaussian. The resulting Gaussian will have a variance that is the sum of the variances of the two initial Gaussians.Let me denote the variance of the initial condition. The initial condition is ( e^{-r^2} ), which can be seen as ( e^{-r^2 / (2 sigma_0^2)} ) with ( sigma_0^2 = 1/2 ). The Green's function has a variance of ( 2 D t ), since ( G ) is ( e^{-r^2 / (4 D t)} ), which is ( e^{-r^2 / (2 sigma^2)} ) with ( sigma^2 = 2 D t ).Therefore, the variance of the convolved function is ( sigma_0^2 + sigma^2 = 1/2 + 2 D t ). Hence, the resulting Gaussian is:[ C(r, t) = frac{C_0}{(1 + 4 D t)^{3/2}} e^{-r^2 / (1 + 4 D t)} ]Wait, let me check the scaling. If the variance is ( 1/2 + 2 D t ), then the exponent should be ( -r^2 / (2 (1/2 + 2 D t)) ), which simplifies to ( -r^2 / (1 + 4 D t) ). So, the exponent is correct.Now, the normalization factor. The initial condition has a factor of ( C_0 ), and the Green's function has a factor of ( 1/(4 pi D t)^{3/2} ). When convolving, the normalization constants multiply, but since we are dealing with probability densities, the total integral should remain constant.Wait, actually, the total concentration should be conserved. Let me compute the integral of ( C(r, t) ) over all space to check.The integral of ( C(r, t) ) is:[ int_{mathbb{R}^3} C(r, t) d^3 r = C_0 int_0^infty frac{1}{(1 + 4 D t)^{3/2}} e^{-r^2 / (1 + 4 D t)} r^2 dr ]Let me make a substitution: let ( u = r^2 / (1 + 4 D t) ), so ( r = sqrt{(1 + 4 D t) u} ), and ( dr = sqrt{(1 + 4 D t)/u} du / 2 ). Wait, maybe it's easier to recall that:[ int_0^infty r^2 e^{-a r^2} dr = frac{sqrt{pi}}{4 a^{3/2}} ]So, substituting ( a = 1/(1 + 4 D t) ), we get:[ int_0^infty r^2 e^{-r^2 / (1 + 4 D t)} dr = frac{sqrt{pi}}{4 (1/(1 + 4 D t))^{3/2}} = frac{sqrt{pi} (1 + 4 D t)^{3/2}}{4} ]Therefore, the integral becomes:[ C_0 times frac{1}{(1 + 4 D t)^{3/2}} times frac{sqrt{pi} (1 + 4 D t)^{3/2}}{4} = C_0 times frac{sqrt{pi}}{4} ]Wait, but the initial integral of ( C(r, 0) ) is:[ int_{mathbb{R}^3} C_0 e^{-r^2} d^3 r = C_0 times frac{sqrt{pi}^3}{(1)^{3/2}}} ]Wait, no. Wait, in 3D, the integral of ( e^{-r^2} ) is ( pi^{3/2} ). So, the initial total concentration is ( C_0 pi^{3/2} ). But according to the solution I found, the integral is ( C_0 sqrt{pi}/4 ), which is different. That can't be right because the total concentration should be conserved.Hmm, so I must have made a mistake in the normalization. Let me go back.The Green's function in 3D is:[ G(mathbf{r}, t) = frac{1}{(4 pi D t)^{3/2}} e^{-r^2 / (4 D t)} ]And the initial condition is:[ C(mathbf{r}, 0) = C_0 e^{-r^2} ]So, the solution is the convolution:[ C(mathbf{r}, t) = int_{mathbb{R}^3} G(mathbf{r} - mathbf{r}', t) C(mathbf{r}', 0) d^3 r' ]Since both are radially symmetric, this becomes:[ C(r, t) = int_0^infty G(r - r') C(r') r'^2 dr' ]But actually, in 3D, the convolution in spherical coordinates is:[ C(r, t) = int_0^infty G(r') C(r'') times text{something} ]Wait, maybe it's better to use the fact that the Fourier transform of the solution is the product of the Fourier transforms of ( G ) and ( C ).So, taking the Fourier transform of both sides of the heat equation:[ mathcal{F}left{frac{partial C}{partial t}right} = mathcal{F}{D nabla^2 C} ]Which gives:[ frac{partial tilde{C}}{partial t} = -D k^2 tilde{C} ]This is an ordinary differential equation in ( t ), with solution:[ tilde{C}(mathbf{k}, t) = tilde{C}(mathbf{k}, 0) e^{-D k^2 t} ]Where ( tilde{C}(mathbf{k}, 0) ) is the Fourier transform of the initial condition.Given ( C(r, 0) = C_0 e^{-r^2} ), its Fourier transform is:[ tilde{C}(mathbf{k}, 0) = C_0 left( frac{pi}{1} right)^{3/2} e^{-pi^2 k^2 / 1} ]Wait, no, earlier I was confused about the constants. Let me recall that the Fourier transform of ( e^{-a r^2} ) in 3D is ( ( pi / a )^{3/2} e^{-pi^2 k^2 / a} ). So, for ( a = 1 ), it's ( pi^{3/2} e^{-pi^2 k^2} ).Therefore,[ tilde{C}(mathbf{k}, t) = C_0 pi^{3/2} e^{-pi^2 k^2} e^{-D k^2 t} = C_0 pi^{3/2} e^{-(pi^2 + D t) k^2} ]Now, to find ( C(r, t) ), we take the inverse Fourier transform:[ C(r, t) = mathcal{F}^{-1}left{ C_0 pi^{3/2} e^{-(pi^2 + D t) k^2} right} ]The inverse Fourier transform of ( e^{-a k^2} ) in 3D is ( frac{1}{(4 pi a)^{3/2}} e^{-r^2 / (4a)} ).So, substituting ( a = pi^2 + D t ), we get:[ C(r, t) = C_0 pi^{3/2} times frac{1}{(4 pi (pi^2 + D t))^{3/2}} e^{-r^2 / (4 (pi^2 + D t))} ]Simplify the constants:First, ( pi^{3/2} times frac{1}{(4 pi (pi^2 + D t))^{3/2}} = frac{pi^{3/2}}{(4 pi)^{3/2} (pi^2 + D t)^{3/2}}} = frac{1}{(4)^{3/2} (pi^2 + D t)^{3/2}}} )Since ( (4 pi)^{3/2} = 4^{3/2} pi^{3/2} ).So,[ C(r, t) = C_0 times frac{1}{(4)^{3/2} (pi^2 + D t)^{3/2}} e^{-r^2 / (4 (pi^2 + D t))} ]But this seems complicated, and I was expecting a simpler expression. Maybe I messed up the constants in the Fourier transform.Alternatively, perhaps it's better to use dimensional analysis or scaling arguments.The heat equation is:[ frac{partial C}{partial t} = D nabla^2 C ]The initial condition is ( C(r, 0) = C_0 e^{-r^2} ). Let's assume the solution has the form:[ C(r, t) = frac{C_0}{(1 + a t)^{3/2}} e^{-r^2 / (1 + a t)} ]We need to find the constant ( a ). Let's plug this into the heat equation.First, compute ( partial C / partial t ):[ frac{partial C}{partial t} = C_0 left[ -frac{3 a}{2} (1 + a t)^{-5/2} e^{-r^2 / (1 + a t)} + (1 + a t)^{-3/2} e^{-r^2 / (1 + a t)} times frac{r^2 a}{(1 + a t)^2} right] ]Simplify:[ frac{partial C}{partial t} = frac{C_0}{(1 + a t)^{5/2}} left[ -frac{3 a}{2} e^{-r^2 / (1 + a t)} + frac{a r^2}{(1 + a t)} e^{-r^2 / (1 + a t)} right] ]Factor out ( e^{-r^2 / (1 + a t)} ):[ frac{partial C}{partial t} = frac{C_0 e^{-r^2 / (1 + a t)}}{(1 + a t)^{5/2}} left( -frac{3 a}{2} + frac{a r^2}{1 + a t} right) ]Now, compute ( nabla^2 C ). Since ( C ) is radially symmetric, ( nabla^2 C = frac{1}{r^2} frac{partial}{partial r} left( r^2 frac{partial C}{partial r} right) ).First, compute ( partial C / partial r ):[ frac{partial C}{partial r} = frac{C_0}{(1 + a t)^{3/2}} e^{-r^2 / (1 + a t)} times left( -frac{2 r}{1 + a t} right) ]So,[ frac{partial C}{partial r} = -frac{2 C_0 r}{(1 + a t)^{5/2}} e^{-r^2 / (1 + a t)} ]Then,[ frac{partial}{partial r} left( r^2 frac{partial C}{partial r} right) = frac{partial}{partial r} left( -frac{2 C_0 r^3}{(1 + a t)^{5/2}} e^{-r^2 / (1 + a t)} right) ]Compute this derivative:Let me denote ( u = r^3 ) and ( v = e^{-r^2 / (1 + a t)} ). Then,[ frac{d}{dr} (u v) = u' v + u v' ]Where,[ u' = 3 r^2 ][ v' = -frac{2 r}{1 + a t} e^{-r^2 / (1 + a t)} ]So,[ frac{d}{dr} (u v) = 3 r^2 e^{-r^2 / (1 + a t)} - frac{2 r^4}{1 + a t} e^{-r^2 / (1 + a t)} ]Therefore,[ frac{partial}{partial r} left( r^2 frac{partial C}{partial r} right) = -frac{2 C_0}{(1 + a t)^{5/2}} left( 3 r^2 - frac{2 r^4}{1 + a t} right) e^{-r^2 / (1 + a t)} ]Thus,[ nabla^2 C = frac{1}{r^2} times left( -frac{2 C_0}{(1 + a t)^{5/2}} left( 3 r^2 - frac{2 r^4}{1 + a t} right) e^{-r^2 / (1 + a t)} right) ]Simplify:[ nabla^2 C = -frac{2 C_0}{(1 + a t)^{5/2} r^2} left( 3 r^2 - frac{2 r^4}{1 + a t} right) e^{-r^2 / (1 + a t)} ][ = -frac{2 C_0}{(1 + a t)^{5/2}} left( 3 - frac{2 r^2}{1 + a t} right) e^{-r^2 / (1 + a t)} ]Now, according to the heat equation:[ frac{partial C}{partial t} = D nabla^2 C ]So, equate the two expressions:[ frac{C_0 e^{-r^2 / (1 + a t)}}{(1 + a t)^{5/2}} left( -frac{3 a}{2} + frac{a r^2}{1 + a t} right) = D times left( -frac{2 C_0}{(1 + a t)^{5/2}} left( 3 - frac{2 r^2}{1 + a t} right) e^{-r^2 / (1 + a t)} right) ]We can cancel out ( frac{C_0 e^{-r^2 / (1 + a t)}}{(1 + a t)^{5/2}} ) from both sides:[ left( -frac{3 a}{2} + frac{a r^2}{1 + a t} right) = D times left( -2 left( 3 - frac{2 r^2}{1 + a t} right) right) ]Simplify the right-hand side:[ -2 D left( 3 - frac{2 r^2}{1 + a t} right) = -6 D + frac{4 D r^2}{1 + a t} ]So, the equation becomes:[ -frac{3 a}{2} + frac{a r^2}{1 + a t} = -6 D + frac{4 D r^2}{1 + a t} ]Now, let's collect like terms. Let's move all terms to the left-hand side:[ -frac{3 a}{2} + frac{a r^2}{1 + a t} + 6 D - frac{4 D r^2}{1 + a t} = 0 ]Factor out ( r^2 ):[ -frac{3 a}{2} + 6 D + left( frac{a}{1 + a t} - frac{4 D}{1 + a t} right) r^2 = 0 ]This equation must hold for all ( r ) and ( t ), so the coefficients of ( r^2 ) and the constant term must each be zero.First, set the coefficient of ( r^2 ) to zero:[ frac{a - 4 D}{1 + a t} = 0 ]This implies ( a - 4 D = 0 ), so ( a = 4 D ).Next, set the constant term to zero:[ -frac{3 a}{2} + 6 D = 0 ]Substitute ( a = 4 D ):[ -frac{3 times 4 D}{2} + 6 D = -6 D + 6 D = 0 ]Which holds true. Therefore, ( a = 4 D ).Thus, the solution is:[ C(r, t) = frac{C_0}{(1 + 4 D t)^{3/2}} e^{-r^2 / (1 + 4 D t)} ]Great, that matches what I initially thought. So, the concentration at any time ( t ) is given by this expression.Now, moving on to the second part: determining the rate of change of the total amount of polymer in a spherical region of radius ( R ) centered at the origin.The total amount ( Q(t) ) is the integral of ( C(r, t) ) over the sphere of radius ( R ):[ Q(t) = int_{V} C(r, t) dV ]Where ( V ) is the volume of the sphere. To find the rate of change, we compute ( dQ/dt ):[ frac{dQ}{dt} = frac{d}{dt} int_{V} C(r, t) dV ]Assuming that the concentration is smooth and the region is fixed (since the sphere is of fixed radius ( R )), we can interchange the derivative and the integral:[ frac{dQ}{dt} = int_{V} frac{partial C}{partial t} dV ]But from the PDE, we know that ( frac{partial C}{partial t} = D nabla^2 C ). Therefore,[ frac{dQ}{dt} = D int_{V} nabla^2 C , dV ]Using the divergence theorem, the integral of the Laplacian over a volume is equal to the integral of the normal derivative over the surface:[ int_{V} nabla^2 C , dV = int_{S} frac{partial C}{partial n} dS ]Where ( S ) is the surface of the sphere, and ( frac{partial C}{partial n} ) is the normal derivative (outward pointing) at the surface.Therefore,[ frac{dQ}{dt} = D int_{S} frac{partial C}{partial n} dS ]Since we're dealing with a sphere of radius ( R ), the normal derivative is just the radial derivative at ( r = R ):[ frac{partial C}{partial n} = frac{partial C}{partial r} bigg|_{r=R} ]So,[ frac{dQ}{dt} = D int_{S} frac{partial C}{partial r} bigg|_{r=R} dS ]The surface area element ( dS ) on a sphere of radius ( R ) is ( R^2 sin theta dtheta dphi ), but since the integrand is radially symmetric (it depends only on ( r )), the integral simplifies to:[ frac{dQ}{dt} = D times frac{partial C}{partial r} bigg|_{r=R} times text{Surface Area} ]The surface area of a sphere is ( 4 pi R^2 ). Therefore,[ frac{dQ}{dt} = D times frac{partial C}{partial r} bigg|_{r=R} times 4 pi R^2 ]Now, let's compute ( frac{partial C}{partial r} ) at ( r = R ).From the solution:[ C(r, t) = frac{C_0}{(1 + 4 D t)^{3/2}} e^{-r^2 / (1 + 4 D t)} ]Compute the radial derivative:[ frac{partial C}{partial r} = frac{C_0}{(1 + 4 D t)^{3/2}} times left( -frac{2 r}{1 + 4 D t} right) e^{-r^2 / (1 + 4 D t)} ]At ( r = R ):[ frac{partial C}{partial r} bigg|_{r=R} = -frac{2 C_0 R}{(1 + 4 D t)^{5/2}} e^{-R^2 / (1 + 4 D t)} ]Therefore,[ frac{dQ}{dt} = D times left( -frac{2 C_0 R}{(1 + 4 D t)^{5/2}} e^{-R^2 / (1 + 4 D t)} right) times 4 pi R^2 ]Simplify:[ frac{dQ}{dt} = -8 pi D C_0 R^3 frac{e^{-R^2 / (1 + 4 D t)}}{(1 + 4 D t)^{5/2}} ]Alternatively, we can write this as:[ frac{dQ}{dt} = -frac{8 pi D C_0 R^3}{(1 + 4 D t)^{5/2}} e^{-R^2 / (1 + 4 D t)} ]This expression gives the rate of change of the total polymer amount in the sphere. The negative sign indicates that the amount is decreasing over time, which makes sense because the polymer is diffusing out of the sphere.Alternatively, if we consider the flux of polymer across the surface, the rate of change is equal to the negative of the flux out of the sphere. The flux is given by Fick's law:[ mathbf{J} = -D nabla C ]So, the outward flux through the surface is:[ int_S mathbf{J} cdot mathbf{n} , dS = -D int_S frac{partial C}{partial n} dS ]Which is exactly the expression we derived for ( dQ/dt ), confirming the result.So, summarizing:1. The concentration at any time ( t ) is:[ C(r, t) = frac{C_0}{(1 + 4 D t)^{3/2}} e^{-r^2 / (1 + 4 D t)} ]2. The rate of change of the total amount of polymer in the sphere is:[ frac{dQ}{dt} = -frac{8 pi D C_0 R^3}{(1 + 4 D t)^{5/2}} e^{-R^2 / (1 + 4 D t)} ]Alternatively, this can be expressed in terms of the initial concentration distribution. Since the initial concentration is ( C_0 e^{-r^2} ), at ( t = 0 ), the concentration at ( r = R ) is ( C_0 e^{-R^2} ). However, in the expression for ( dQ/dt ), it's already in terms of ( C_0 ), so perhaps that's sufficient.But wait, the problem says "express this rate of change in terms of the diffusion coefficient ( D ) and the initial concentration distribution." So, maybe we can write it using the initial concentration at ( r = R ).Let me denote ( C(R, 0) = C_0 e^{-R^2} ). Then, ( C_0 = C(R, 0) e^{R^2} ). Substituting back into the expression for ( dQ/dt ):[ frac{dQ}{dt} = -frac{8 pi D (C(R, 0) e^{R^2}) R^3}{(1 + 4 D t)^{5/2}} e^{-R^2 / (1 + 4 D t)} ]Simplify the exponent:[ e^{R^2} times e^{-R^2 / (1 + 4 D t)} = e^{R^2 (1 - 1/(1 + 4 D t))} = e^{R^2 (4 D t / (1 + 4 D t))} ]So,[ frac{dQ}{dt} = -8 pi D C(R, 0) R^3 frac{e^{4 D t R^2 / (1 + 4 D t)}}{(1 + 4 D t)^{5/2}} ]But this might not be necessary unless specifically asked. The original expression in terms of ( C_0 ) is probably acceptable.Alternatively, another approach to find ( dQ/dt ) is to use the continuity equation. The rate of change of concentration is related to the flux. Since the total amount is ( Q(t) = int_V C dV ), then:[ frac{dQ}{dt} = int_V frac{partial C}{partial t} dV = int_V D nabla^2 C dV = D int_S frac{partial C}{partial n} dS ]Which is the same as before. So, this confirms the result.Therefore, Raj can use these results to analyze his polymer diffusion project.</think>