🧠:<think>Okay, so I've got this problem about a legal contract that involves some numerical guidelines. It's split into two parts, and I need to tackle them one by one. Let me start by understanding what each part is asking.First, the proofreader needs to ensure that the total amount of monetary transactions, represented by ( T ), is evenly distributed over 12 months. Each monthly transaction must be an integer, and ( T ) itself must also be an integer. The goal is to find the smallest possible ( T ) such that each monthly transaction amount is a prime number.Alright, so breaking this down: ( T ) is the total amount, and it's divided equally over 12 months. So each month's transaction is ( T / 12 ). Since both ( T ) and each monthly amount need to be integers, ( T ) must be divisible by 12. Also, ( T / 12 ) has to be a prime number.So, the first part is essentially asking for the smallest integer ( T ) such that ( T ) is divisible by 12 and ( T / 12 ) is prime. That makes sense. So, the smallest prime number is 2, right? So if I take ( T / 12 = 2 ), then ( T = 24 ). Is 24 divisible by 12? Yes, it is. So, is 24 the smallest possible ( T )?Wait, hold on. Let me think again. The problem says each monthly transaction must be a prime number. So, if ( T / 12 ) is prime, then ( T ) is 12 times a prime number. So, the smallest prime is 2, so ( T = 24 ). That seems straightforward.But let me double-check. If ( T = 24 ), then each month's transaction is 2, which is prime. So, that should be the smallest ( T ). I think that's correct.Moving on to the second part. The contract also requires that the sum of any two consecutive monthly transactions is a perfect square. Given the smallest ( T ) found in part 1, which is 24, we need to verify if this condition holds. If not, we have to find the next smallest ( T ) that satisfies both conditions.Wait, hold on. If each monthly transaction is the same, because ( T ) is evenly distributed, then each month's transaction is 2. So, the sum of any two consecutive transactions would be 2 + 2 = 4, which is a perfect square (2 squared). So, actually, the condition is satisfied.But wait, that seems too easy. Maybe I'm misunderstanding the problem. Let me reread it."The sum of any two consecutive monthly transactions is a perfect square." Hmm, so if each transaction is the same, then yes, the sum is 4, which is a square. So, with ( T = 24 ), each transaction is 2, and the sum of any two consecutive is 4, which is a square. So, both conditions are satisfied.But that seems too straightforward. Maybe I'm missing something. Let me think again.Wait, perhaps the problem is more complex. Maybe the transactions aren't all the same? But the problem says the total amount must be evenly distributed over 12 months. So, each month's transaction is equal. So, each is ( T / 12 ). So, if ( T / 12 ) is prime, then each transaction is a prime number, and the sum of any two consecutive is twice that prime.So, for the sum to be a perfect square, twice the prime must be a perfect square. So, in the case of ( T = 24 ), each transaction is 2, so the sum is 4, which is a perfect square. So, that works.But wait, if I take the next prime, which is 3. Then ( T = 36 ). Each transaction is 3, so the sum is 6, which is not a perfect square. So, that doesn't work. Next prime is 5, ( T = 60 ), sum is 10, not a square. Next prime is 7, sum is 14, not a square. Next prime is 11, sum is 22, not a square. Next prime is 13, sum is 26, not a square. Next prime is 17, sum is 34, not a square. Next prime is 19, sum is 38, not a square. Next prime is 23, sum is 46, not a square. Next prime is 29, sum is 58, not a square. Next prime is 31, sum is 62, not a square. Next prime is 37, sum is 74, not a square. Next prime is 41, sum is 82, not a square. Next prime is 43, sum is 86, not a square. Next prime is 47, sum is 94, not a square.Wait a minute, so the only prime where twice the prime is a perfect square is when the prime is 2, because 2*2=4, which is 2 squared. All other primes, when doubled, don't result in a perfect square. So, that means the only possible ( T ) that satisfies both conditions is 24.But wait, the problem says "verify if this condition holds true. If not, find the next smallest ( T ) that satisfies both conditions." So, in this case, since the condition does hold true for ( T = 24 ), we don't need to look for a larger ( T ).But that seems too straightforward. Maybe I'm misinterpreting the problem. Let me read it again."1. The contract states that the total amount of monetary transactions must be evenly distributed over a period of 12 months. The total transaction amount is represented by ( T ). If ( T ) is an integer and each monthly transaction amount must be an integer as well, determine the smallest possible ( T ) such that the monthly transaction amount is a prime number."Okay, so ( T ) must be divisible by 12, and ( T / 12 ) must be prime. So, the smallest ( T ) is 24."2. The contract also stipulates that the total transaction amount ( T ) must satisfy the condition that the sum of any two consecutive monthly transactions is a perfect square. Given the smallest ( T ) found in sub-problem 1, verify if this condition holds true. If not, find the next smallest ( T ) that satisfies both conditions."So, if ( T = 24 ), each transaction is 2, so the sum of any two consecutive is 4, which is a perfect square. So, condition holds. Therefore, ( T = 24 ) satisfies both conditions.But wait, maybe the problem is more complex. Maybe the transactions aren't all the same? But the problem says "evenly distributed," which I took to mean equal each month. But perhaps "evenly distributed" could mean something else, like spread out in a way that each month's amount is the same, which would mean equal. So, I think my initial interpretation is correct.Alternatively, maybe "evenly distributed" could mean that the amounts are spread out in a way that the sum of any two consecutive is a square, but that seems to be the second condition. So, perhaps the first condition is just that ( T ) is divisible by 12, and each monthly amount is prime, but not necessarily equal? Wait, no, the problem says "evenly distributed," which I think implies equal.Wait, let me check the exact wording: "the total amount of monetary transactions must be evenly distributed over a period of 12 months." So, "evenly distributed" usually means equal amounts each period. So, each month's transaction is ( T / 12 ). So, that must be an integer, and also a prime number.So, in that case, ( T = 12 times p ), where ( p ) is prime. The smallest such ( T ) is 24, as 2 is the smallest prime.Then, for the second condition, the sum of any two consecutive monthly transactions must be a perfect square. Since all monthly transactions are equal, the sum is ( 2p ). So, ( 2p ) must be a perfect square.So, ( 2p = k^2 ), where ( k ) is an integer. Therefore, ( p = k^2 / 2 ). Since ( p ) is prime, ( k^2 ) must be even, so ( k ) must be even. Let ( k = 2m ), then ( p = (4m^2)/2 = 2m^2 ). But ( p ) is prime, so ( 2m^2 ) must be prime. The only way that happens is if ( m = 1 ), because if ( m > 1 ), ( m^2 ) is greater than 1, so ( 2m^2 ) would be composite. So, ( m = 1 ), ( k = 2 ), ( p = 2 ). Therefore, ( T = 24 ).So, that's the only possible ( T ) that satisfies both conditions. Therefore, the answer is 24.But wait, let me think again. Is there a way for the monthly transactions to not be equal but still have ( T ) divisible by 12 and each monthly amount prime, and the sum of any two consecutive being a perfect square? Because if that's possible, maybe there's a smaller ( T ) than 24? But the problem says "evenly distributed," which I think implies equal. So, I think my initial conclusion is correct.Alternatively, if "evenly distributed" doesn't necessarily mean equal, but just that the distribution is even, perhaps in some other way. But I think in the context of monetary transactions, "evenly distributed" would mean equal amounts each month. So, I think my answer is correct.Therefore, the smallest ( T ) is 24, and it satisfies both conditions.</think>

🧠:<think>Okay, so I've got this problem about an environmental scientist analyzing a city's architectural design. It's split into two parts. Let me try to tackle them one by one.Starting with the first part: we have a grid of buildings with dimensions n x m. Each building's height is given by h_{i,j} = a sin(i) + b cos(j). The sun is at a zenith angle θ, and it's casting shadows only along the x-axis. We need to find the total shadow area S.Hmm, okay. So, the sun's angle is θ, which is the angle from the vertical. So, the sun is at an elevation angle of θ, meaning it's above the horizon by θ. Since shadows are cast along the x-axis, that probably means the sun is coming from the positive x-direction, casting shadows towards the negative x-direction.To find the shadow area, I think we need to figure out for each building, how much area it shades on the ground. Since the sun is at an angle θ, the length of the shadow each building casts will depend on its height and the angle.Wait, but the grid is 2D, so each building is at a position (i, j). But the shadow is cast along the x-axis, so each building will cast a shadow to the left (negative x-direction). The shadow length for each building would be h_{i,j} / tan(θ), right? Because tan(θ) = opposite/adjacent, so adjacent (shadow length) = opposite (height) / tan(θ).But wait, actually, if θ is the zenith angle, which is the angle from the vertical, then the elevation angle is 90° - θ. So, the sun's angle above the horizon is 90° - θ. So, the shadow length would be h_{i,j} / tan(90° - θ). But tan(90° - θ) is cot(θ), so the shadow length is h_{i,j} * tan(θ). Wait, let me double-check.If θ is the angle from the vertical, then the elevation angle is 90° - θ. So, the shadow length is h / tan(elevation angle). So, that would be h / tan(90° - θ) = h * tan(θ). Yeah, that seems right.So, each building at (i, j) will cast a shadow of length h_{i,j} * tan(θ) along the negative x-axis. So, the shadow will start at x = i - h_{i,j} * tan(θ) and extend to x = i. But wait, the grid is discrete, so each building is at integer coordinates? Or is it continuous?Wait, the grid is n x m, but the problem doesn't specify whether i and j are integers or real numbers. Hmm. It just says a grid, so maybe they are discrete points. But the height function is defined for each (i, j), so perhaps it's a continuous grid? Or maybe it's a discrete grid where each cell is a building.Wait, the problem says "each building has a height h_{i,j}", so I think it's a grid where each cell is a building, so i and j are integers from 1 to n and 1 to m, respectively.But then, if the grid is discrete, how do we calculate the shadow area? Because each building is a single point? Or is each building a square of unit area?Wait, the problem says "the city is modeled as a rectangular grid of buildings", so perhaps each building is a square with unit dimensions? So, each building is 1x1 square at position (i, j). So, the shadow cast by each building would be a rectangle extending to the left from (i, j) by length h_{i,j} * tan(θ). But since the grid is 2D, the shadow would overlap with other buildings or ground.But wait, the shadow area is the area on the ground that is shaded. So, for each building, the shadow it casts is a rectangle of width 1 (since each building is 1 unit wide) and length h_{i,j} * tan(θ). But if the shadow extends beyond the grid, we might have to consider that.But actually, since the sun is casting shadows along the x-axis, each building at (i, j) will cast a shadow from x = i - h_{i,j} * tan(θ) to x = i, and y from j - 0.5 to j + 0.5 (assuming each building is 1 unit wide in y-direction). But wait, the grid is n x m, so perhaps the buildings are arranged in a grid where each is 1 unit apart? Or is it a continuous grid?This is a bit confusing. Maybe I need to make some assumptions. Let's assume that each building is a unit square at position (i, j), so the base is from (i, j) to (i+1, j+1). Then, the shadow cast by each building would extend to the left along the x-axis. So, the shadow would start at x = i - h_{i,j} * tan(θ) and end at x = i, and in the y-direction, it would cover from j to j+1.But wait, the shadow might overlap with other buildings. So, the total shadow area isn't just the sum of all individual shadows because overlapping areas would be double-counted. So, this complicates things.Alternatively, maybe the grid is such that each building is a point, and the shadow is a line segment. But then the shadow area would be zero, which doesn't make sense.Wait, maybe the grid is a 2D plane, and each building is a vertical column at position (i, j) with height h_{i,j}. So, the shadow of each building is a rectangle on the ground, extending from (i - h_{i,j} * tan(θ), j - 0.5) to (i, j + 0.5), assuming each building has a width of 1 unit in the y-direction.But if that's the case, then the shadow area for each building is 1 * h_{i,j} * tan(θ). So, the total shadow area S would be the sum over all i and j of h_{i,j} * tan(θ). But we have to be careful about overlapping shadows.Wait, but if the buildings are close together, their shadows might overlap. So, the total shadow area isn't just the sum of individual shadows because overlapping regions are counted multiple times. So, to get the actual shadow area, we need to consider the union of all shadows, which is more complicated.But the problem says "derive an expression for the total shadow area S". It doesn't specify whether overlapping shadows should be considered or not. Maybe it's assuming that shadows don't overlap, which would be the case if the buildings are spaced far apart. But in a city grid, buildings are usually adjacent or close, so shadows might overlap.Alternatively, maybe the problem is simplifying and just wants the sum of all individual shadows, regardless of overlap. So, S = tan(θ) * sum_{i=1 to n} sum_{j=1 to m} h_{i,j}.Given that h_{i,j} = a sin(i) + b cos(j), then S = tan(θ) * sum_{i=1 to n} sum_{j=1 to m} [a sin(i) + b cos(j)].We can separate the sums:S = tan(θ) * [a sum_{i=1 to n} sin(i) * m + b sum_{j=1 to m} cos(j) * n]Because for each i, sin(i) is multiplied by m terms in j, and for each j, cos(j) is multiplied by n terms in i.So, S = tan(θ) * [a m sum_{i=1 to n} sin(i) + b n sum_{j=1 to m} cos(j)].Now, we can compute these sums. The sum of sin(i) from i=1 to n is a known trigonometric series. Similarly for cos(j).Recall that sum_{k=1 to N} sin(k) = [sin(N/2) * sin((N+1)/2)] / sin(1/2). Similarly, sum_{k=1 to N} cos(k) = [sin(N/2) * cos((N+1)/2)] / sin(1/2).So, applying these formulas:sum_{i=1 to n} sin(i) = [sin(n/2) * sin((n+1)/2)] / sin(1/2)sum_{j=1 to m} cos(j) = [sin(m/2) * cos((m+1)/2)] / sin(1/2)Therefore, substituting back into S:S = tan(θ) * [a m * [sin(n/2) sin((n+1)/2) / sin(1/2)] + b n * [sin(m/2) cos((m+1)/2) / sin(1/2)] ]We can factor out 1 / sin(1/2):S = [tan(θ) / sin(1/2)] * [a m sin(n/2) sin((n+1)/2) + b n sin(m/2) cos((m+1)/2)]That's the expression for the total shadow area S.Wait, but I'm assuming that each building's shadow doesn't overlap with others. If they do overlap, this expression would overcount the shadow area. However, since the problem doesn't specify handling overlaps, maybe this is the intended answer.Moving on to the second part: the scientist considers the effect of building heights on wind flow. Using h_{i,j} = a sin(i) + b cos(j), we need to find the critical points where the gradient ∇h is zero. These points might correspond to regions of low wind velocity.So, the gradient of h is a vector with partial derivatives in x and y directions. But in our case, the grid is 2D with i and j as indices. Assuming i is the x-direction and j is the y-direction, then the gradient would be (∂h/∂i, ∂h/∂j).But since h is a function of i and j, which are discrete indices, we might need to consider the gradient in the continuous sense. Alternatively, if we treat i and j as continuous variables, then we can take partial derivatives.So, treating i and j as continuous variables, h(i, j) = a sin(i) + b cos(j).Then, ∂h/∂i = a cos(i)∂h/∂j = -b sin(j)So, the gradient ∇h = (a cos(i), -b sin(j))To find critical points, set ∇h = 0, so:a cos(i) = 0-b sin(j) = 0So, cos(i) = 0 and sin(j) = 0.Solutions for cos(i) = 0 are i = π/2 + kπ, where k is integer.Solutions for sin(j) = 0 are j = lπ, where l is integer.Therefore, critical points occur at (i, j) = (π/2 + kπ, lπ).These points are where the height function has local maxima, minima, or saddle points.In terms of wind flow, regions where the gradient is zero might correspond to areas where the wind velocity is low because wind tends to follow the pressure gradient, which is related to the height gradient in the atmosphere. So, if the gradient is zero, there's no driving force for wind, leading to calm areas. These regions could trap pollutants, affecting air quality.But wait, in reality, wind is influenced by many factors, including pressure gradients, Coriolis effect, etc., but in this simplified model, assuming wind follows the height gradient, then yes, critical points would be areas of low wind velocity.So, summarizing, the critical points are at (π/2 + kπ, lπ), and these could be regions where wind velocity is low, impacting air quality.I think that's the gist of it. Let me just recap:1. For the shadow area, assuming no overlap, S is the sum of all individual shadows, which is tan(θ) times the sum of h_{i,j} over all i and j. Using the trigonometric sums, we get the expression involving sine and cosine terms.2. For the critical points, solving ∇h = 0 gives specific (i, j) positions where the height function has extrema or saddle points, which could correspond to low wind areas.I hope I didn't make any mistakes in the trigonometric sums. Let me double-check the sum formulas.Yes, the sum of sin(k) from k=1 to N is indeed [sin(N/2) sin((N+1)/2)] / sin(1/2). Similarly for cosine. So, that part seems correct.And for the critical points, setting the partial derivatives to zero gives the correct solutions. So, that should be fine.</think>

🧠:<think>First, I need to determine the total number of square photo frames required to cover the perimeter of the room. The room is rectangular, measuring 20 meters in length and 15 meters in width. Each photo frame has a side length of 1 meter.To find the perimeter of the room, I'll use the formula for the perimeter of a rectangle:Perimeter = 2 × (Length + Width)Plugging in the values:Perimeter = 2 × (20 meters + 15 meters) = 2 × 35 meters = 70 metersSince each frame is 1 meter in length, the total number of frames needed to cover the entire perimeter is equal to the perimeter measurement. Therefore, 70 frames are required.Next, the mother wants to alternate between black and white frames, starting with a black frame at one corner. To determine the number of black and white frames, I'll consider the total number of frames and the alternating pattern.If the pattern starts with black, the sequence will be black, white, black, white, and so on. Since the total number of frames is 70, which is an even number, the number of black and white frames will be equal.Calculating the number of black frames:Number of black frames = Total frames / 2 = 70 / 2 = 35Similarly, the number of white frames will also be:Number of white frames = 70 / 2 = 35So, there will be 35 black frames and 35 white frames along the perimeter.</think>

🧠:<think>Alright, so I have this problem about a congresswoman analyzing a new healthcare policy. It's split into two parts, and I need to figure out the relationships between the constants in the cost functions and then formulate an equation for the percentage reduction in cost. Let me take it step by step.Starting with part 1. The current cost function is a quadratic: ( C(x) = ax^2 + bx + c ). The new policy changes this to ( C'(x) = k log(x + 1) + mx + n ). They mention that ( C(0) = C'(0) ), so that gives me one equation. Also, the rate of cost reduction at ( x = 50% ) is twice as fast under the new policy. Hmm, rate of cost reduction would be the derivative, right? So I need to take the derivatives of both functions at ( x = 50 ) and set the derivative of the new function equal to twice the derivative of the old one.Let me write down what I know:1. ( C(0) = C'(0) )2. ( C'(50) = 2 cdot C(50) ) (Wait, no. It's the rate of cost reduction, so it's the derivative. So actually, ( C'(50) = 2 cdot C(50) )? Wait, no. Wait, the rate of cost reduction is the derivative, so the derivative of C'(x) at x=50 is twice the derivative of C(x) at x=50. So, ( C'(50) = 2 cdot C(50) ). Wait, no, the derivative of C'(x) is ( C''(x) )? No, wait, C'(x) is the new function, so its derivative is ( C''(x) ). Wait, no, wait, in calculus, the derivative of C(x) is C’(x), but here C’(x) is the new function. So maybe I should denote the derivatives differently to avoid confusion.Let me clarify:Let’s denote the derivative of C(x) as ( C'(x) ) and the derivative of C’(x) as ( C''(x) ). So, the rate of cost reduction under the current policy is ( C'(x) = 2ax + b ). Under the new policy, the rate of cost reduction is ( C''(x) = frac{k}{x + 1} + m ).Given that at x=50, the new policy's rate is twice as fast. So:( C''(50) = 2 cdot C'(50) )Which translates to:( frac{k}{50 + 1} + m = 2 cdot (2a cdot 50 + b) )Simplify that:( frac{k}{51} + m = 2(100a + b) )( frac{k}{51} + m = 200a + 2b )That's one equation.Also, since ( C(0) = C'(0) ), let's compute both:( C(0) = a(0)^2 + b(0) + c = c )( C'(0) = k log(0 + 1) + m(0) + n = k log(1) + 0 + n = 0 + n = n )So, ( c = n ). That's another equation.So far, I have two equations:1. ( frac{k}{51} + m = 200a + 2b )2. ( c = n )But I need to find the relationship between a, b, c and k, m, n. So, I have two equations but three variables on each side. Maybe there's another condition? The problem doesn't specify any other points or conditions. Wait, maybe I need to consider that the cost functions are supposed to align at some other point or something else? Hmm, the problem only mentions C(0) = C'(0) and the derivative condition at x=50. So, perhaps that's all.So, the relationships are:- ( n = c )- ( frac{k}{51} + m = 200a + 2b )So, that's the relationship between the constants.Moving on to part 2. The congresswoman aims for a 20% reduction in overall costs when x=60%. So, the new cost at x=60 should be 80% of the original cost at x=60.So, ( C'(60) = 0.8 cdot C(60) )We need to find the percentage reduction in cost per unit percentage increase in preventive care utilization from x=50% to x=60%. So, that would be the change in cost divided by the change in x, but expressed as a percentage.Wait, percentage reduction per unit percentage increase. So, maybe the average rate of change from 50 to 60, expressed as a percentage.Alternatively, it could be the derivative at some point, but the question says "percentage reduction in cost per unit percentage increase", so it's the change in cost divided by the change in x, multiplied by 100 to get a percentage.So, from x=50 to x=60, the change in x is 10 percentage points. The change in cost is ( C'(60) - C'(50) ). But since it's a reduction, it's ( C'(50) - C'(60) ). So, the percentage reduction per unit percentage increase would be ( frac{C'(50) - C'(60)}{10} times 100% ).But wait, the question says "formulate an equation to find the percentage reduction in cost per unit percentage increase in preventive care utilization from x=50% to x=60% under the new policy."So, maybe it's the average rate of change over that interval, expressed as a percentage. So, the formula would be:( text{Percentage Reduction} = frac{C'(50) - C'(60)}{60 - 50} times 100% )Simplify:( text{Percentage Reduction} = frac{C'(50) - C'(60)}{10} times 100% )Which is:( text{Percentage Reduction} = 10 times (C'(50) - C'(60)) % )But since ( C'(x) = k log(x + 1) + mx + n ), we can plug that in:( C'(50) = k log(51) + 50m + n )( C'(60) = k log(61) + 60m + n )So, the difference:( C'(50) - C'(60) = k (log(51) - log(61)) + 50m - 60m + n - n )Simplify:( = k logleft(frac{51}{61}right) - 10m )So, the percentage reduction is:( 10 times (k log(51/61) - 10m) % )Wait, no. Wait, the difference is ( C'(50) - C'(60) = k (log 51 - log 61) - 10m ). So, the percentage reduction is:( frac{C'(50) - C'(60)}{10} times 100% = 10 times (C'(50) - C'(60)) % )Wait, no, the percentage reduction is:( frac{text{Reduction}}{text{Original Cost}} times 100% ). Wait, no, the question says "percentage reduction in cost per unit percentage increase". Hmm, maybe I misinterpreted.Wait, let's read it again: "formulate an equation to find the percentage reduction in cost per unit percentage increase in preventive care utilization from x=50% to x=60% under the new policy."So, it's the percentage reduction in cost divided by the percentage increase in x, which is 10 percentage points. So, it's (C'(50) - C'(60)) / (60 - 50) * 100%.So, yes, that's what I had earlier: 10*(C'(50) - C'(60))%.But since we have expressions for C'(50) and C'(60), we can write:Percentage Reduction = 10*(k log(51) + 50m + n - (k log(61) + 60m + n))% = 10*(k (log51 - log61) -10m)% = 10k (log51 - log61) - 100m)% = 10k log(51/61) - 100m)%.So, the equation is:Percentage Reduction = [10k log(51/61) - 100m] %Now, to analyze the sensitivity with respect to m and k. That means, how does the percentage reduction change as m or k change.Looking at the equation, the percentage reduction is linear in both m and k. The coefficient for k is 10 log(51/61), which is a negative number because 51/61 <1, so log is negative. Similarly, the coefficient for m is -100. So, increasing k would decrease the percentage reduction (since it's multiplied by a negative), and increasing m would also decrease the percentage reduction.Wait, but percentage reduction is a positive quantity, right? So, if the coefficient is negative, increasing k would make the percentage reduction smaller, meaning the policy is less effective. Similarly, increasing m would also make the percentage reduction smaller.So, the sensitivity is such that both k and m have negative impacts on the percentage reduction. The sensitivity with respect to k is 10 log(51/61), which is approximately 10 * (-0.189) ≈ -1.89. So, for each unit increase in k, the percentage reduction decreases by about 1.89%.The sensitivity with respect to m is -100, meaning for each unit increase in m, the percentage reduction decreases by 100%.So, m has a much larger impact on the percentage reduction than k. Therefore, the policy's effectiveness (percentage reduction) is more sensitive to changes in m than in k.Wait, but let me double-check. The percentage reduction is [10k log(51/61) - 100m]%. So, if k increases by Δk, the percentage reduction changes by 10 log(51/61) * Δk, which is negative. Similarly, if m increases by Δm, the percentage reduction changes by -100 Δm.So, the sensitivity (derivative) with respect to k is 10 log(51/61) ≈ -1.89, and with respect to m is -100. So, yes, m has a much larger sensitivity.Therefore, the percentage reduction is more sensitive to changes in m than in k.So, summarizing part 2:The equation for the percentage reduction is:Percentage Reduction = [10k log(51/61) - 100m] %And the sensitivity analysis shows that the reduction is more sensitive to m, with a sensitivity of -100, compared to k's sensitivity of approximately -1.89.I think that's it. Let me just recap:For part 1, the relationships are n = c and (k/51) + m = 200a + 2b.For part 2, the percentage reduction equation is as above, and sensitivity is higher for m.</think>