🧠:<think>Alright, so I have this research project to work on, and I need to tackle two main parts. The first part is about modeling population growth using a logistic function, and the second part is analyzing the socio-economic impact using a linear regression model. Let me take it step by step.Starting with the first problem: the population growth model. The formula given is the logistic growth function:[ P(t) = frac{K}{1 + frac{K - P_0}{P_0} e^{-rt}} ]I need to find the population after 10 years, so t is 10. The given values are:- Initial population, ( P_0 = 500 )- Carrying capacity, ( K = 50,000 )- Growth rate, ( r = 0.05 )Okay, so plugging these into the formula. Let me write it out:[ P(10) = frac{50,000}{1 + frac{50,000 - 500}{500} e^{-0.05 times 10}} ]First, let me compute the denominator step by step. The term ( frac{50,000 - 500}{500} ) is:[ frac{49,500}{500} = 99 ]So the denominator becomes:[ 1 + 99 e^{-0.05 times 10} ]Calculating the exponent:[ -0.05 times 10 = -0.5 ]So now, I need to compute ( e^{-0.5} ). I remember that ( e^{-0.5} ) is approximately 0.6065. Let me double-check that with a calculator to be precise. Yes, ( e^{-0.5} ) is roughly 0.6065.So plugging that back in:[ 1 + 99 times 0.6065 ]Calculating 99 * 0.6065:First, 100 * 0.6065 is 60.65, so subtracting 0.6065 gives 60.65 - 0.6065 = 60.0435.So the denominator is:[ 1 + 60.0435 = 61.0435 ]Now, the entire population formula becomes:[ P(10) = frac{50,000}{61.0435} ]Calculating that division:50,000 divided by 61.0435. Let me do this step by step.First, 61.0435 * 800 = 48,834.8Subtracting that from 50,000: 50,000 - 48,834.8 = 1,165.2Now, 61.0435 * 19 = approximately 1,159.8265So 800 + 19 = 819, and the remainder is 1,165.2 - 1,159.8265 ≈ 5.3735So approximately, 50,000 / 61.0435 ≈ 819.19So rounding to the nearest whole number, that's approximately 819 people.Wait, let me verify this calculation because 61.0435 * 819 is roughly 50,000?61.0435 * 800 = 48,834.861.0435 * 19 = 1,159.8265Adding them together: 48,834.8 + 1,159.8265 ≈ 49,994.6265, which is very close to 50,000. So 819 gives us about 49,994.63, so 819 is a good approximation, but since the exact division would be slightly more than 819, maybe 819.19.But since population can't be a fraction, we can round it to 819 or 820. Let me check 61.0435 * 819.19:61.0435 * 800 = 48,834.861.0435 * 19.19 ≈ 61.0435 * 20 = 1,220.87, subtract 61.0435 * 0.81 ≈ 49.52, so 1,220.87 - 49.52 ≈ 1,171.35Adding to 48,834.8: 48,834.8 + 1,171.35 ≈ 49,996.15, which is still a bit less than 50,000. So maybe 819.2.But for the purposes of this problem, I think 819 is a reasonable estimate.Wait, actually, let me use a calculator for more precision.50,000 divided by 61.0435.Compute 50,000 / 61.0435:Let me compute 61.0435 * 819 = ?61.0435 * 800 = 48,834.861.0435 * 19 = 1,159.8265Total: 48,834.8 + 1,159.8265 = 49,994.6265So 819 gives 49,994.6265, which is 5.3735 less than 50,000.So 5.3735 / 61.0435 ≈ 0.088.So total is approximately 819.088, which is about 819.09.So approximately 819.09, which we can round to 819.1 or 819 people.But since population is in whole numbers, 819 is the closest whole number.So, after 10 years, the population is approximately 819 Namibian Americans.Wait, but let me think again. The formula is:[ P(t) = frac{K}{1 + frac{K - P_0}{P_0} e^{-rt}} ]So plugging in the numbers:K = 50,000P0 = 500r = 0.05t = 10So:Denominator: 1 + (50,000 - 500)/500 * e^(-0.05*10)Which is 1 + (49,500 / 500) * e^(-0.5)49,500 / 500 is 99, as I had before.e^(-0.5) is approximately 0.6065.So 99 * 0.6065 ≈ 60.0435So denominator is 1 + 60.0435 = 61.0435So P(10) = 50,000 / 61.0435 ≈ 819.09, so 819.Yes, that seems correct.Moving on to the second part: the socio-economic impact analysis. I need to determine the constants a and b in the linear regression model:[ I = aN + b ]Given the data points:- When N = 1000, I = 45,000- When N = 2000, I = 50,000- When N = 3000, I = 55,000So, we have three points: (1000, 45000), (2000, 50000), (3000, 55000)I need to find the best fit line for these points. Since it's a linear regression, I can use the least squares method.But since there are only three points, and they seem to form a straight line, maybe the line passes through all three points exactly. Let me check.Compute the slope between the first two points:Slope (a) = (50,000 - 45,000) / (2000 - 1000) = 5,000 / 1,000 = 5.Now, check the slope between the second and third points:(55,000 - 50,000) / (3000 - 2000) = 5,000 / 1,000 = 5.So the slope is consistent. Therefore, the line passes through all three points with a slope of 5.Now, to find the intercept b, we can use any of the points. Let's use the first point (1000, 45000):45,000 = 5 * 1000 + b45,000 = 5,000 + bSo, b = 45,000 - 5,000 = 40,000.Therefore, the equation is:I = 5N + 40,000.Let me verify with the other points.For N = 2000:I = 5*2000 + 40,000 = 10,000 + 40,000 = 50,000. Correct.For N = 3000:I = 5*3000 + 40,000 = 15,000 + 40,000 = 55,000. Correct.So, yes, the line perfectly fits all three points with a slope of 5 and intercept of 40,000.Therefore, the constants are a = 5 and b = 40,000.Wait, but just to be thorough, let me recall the formula for linear regression.The slope a is given by:a = (nΣ(xy) - ΣxΣy) / (nΣx² - (Σx)²)And the intercept b is:b = (Σy - aΣx) / nWhere n is the number of data points.Let me compute this to confirm.Given:x: 1000, 2000, 3000y: 45000, 50000, 55000Compute Σx, Σy, Σxy, Σx².Σx = 1000 + 2000 + 3000 = 6000Σy = 45000 + 50000 + 55000 = 150,000Σxy:(1000 * 45000) + (2000 * 50000) + (3000 * 55000)= 45,000,000 + 100,000,000 + 165,000,000 = 310,000,000Σx²:1000² + 2000² + 3000² = 1,000,000 + 4,000,000 + 9,000,000 = 14,000,000n = 3Now, compute a:a = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)Plugging in the numbers:Numerator: 3*310,000,000 - 6000*150,000= 930,000,000 - 900,000,000 = 30,000,000Denominator: 3*14,000,000 - (6000)^2= 42,000,000 - 36,000,000 = 6,000,000So a = 30,000,000 / 6,000,000 = 5Which matches our earlier calculation.Now, compute b:b = (Σy - aΣx) / n= (150,000 - 5*6000) / 3= (150,000 - 30,000) / 3= 120,000 / 3 = 40,000Again, matches our previous result.So, yes, the constants are a = 5 and b = 40,000.Therefore, the linear regression model is I = 5N + 40,000.So, summarizing:1. After 10 years, the population is approximately 819 Namibian Americans.2. The linear regression model has a slope of 5 and intercept of 40,000, so I = 5N + 40,000.I think that's all. Let me just recap to make sure I didn't miss anything.For the first part, using the logistic model, plugged in the values correctly, computed the exponent, then the denominator, then divided K by that to get P(10). Got approximately 819.For the second part, noticed that the points lie on a straight line with slope 5, then confirmed using the regression formulas, which gave the same result. So, confident in both answers.Final Answer1. The population after 10 years is boxed{819}.2. The constants are ( a = boxed{5} ) and ( b = boxed{40000} ).</think>

🧠:<think>Alright, so I have this problem to solve, and it's about bioinformatics. Let me try to break it down step by step. First, the problem is divided into two parts. The first part is about statistics, specifically deriving the maximum likelihood estimate (MLE) for a parameter lambda in a Poisson distribution. The second part is about computational complexity and estimating the time required for a motif search algorithm. Let me tackle them one by one.Starting with the first part: We have N genetic sequences, each of length L. We're looking for a motif of length k that appears frequently. For each sequence i, M(i) is the number of times the motif appears, and it follows a Poisson distribution with parameter lambda. We need to find the MLE of lambda based on the observed counts M1, M2, ..., MN.Okay, so I remember that the Poisson distribution is used to model the number of events occurring in a fixed interval of time or space. The probability mass function is P(M = m) = (lambda^m * e^(-lambda)) / m! where lambda is the average rate (mean) of occurrence.For maximum likelihood estimation, we need to find the value of lambda that maximizes the likelihood of observing the data we have. The likelihood function is the product of the probabilities of each observation given the parameter. Since the Poisson distribution is involved, the likelihood function L(lambda) would be the product from i=1 to N of [ (lambda^{M_i} * e^{-lambda}) / M_i! ].But since the factorial terms are constants with respect to lambda, they don't affect the maximization. So, we can focus on maximizing the product of lambda^{M_i} * e^{-lambda} for each i. Taking the natural logarithm of the likelihood function (which is a common step to simplify the maximization) gives us the log-likelihood function:log L(lambda) = sum_{i=1}^N [ M_i * log(lambda) - lambda - log(M_i!) ]Again, the log(M_i!) terms are constants with respect to lambda, so we can ignore them for the purpose of maximization. Thus, the log-likelihood simplifies to:log L(lambda) = sum_{i=1}^N [ M_i * log(lambda) - lambda ]To find the MLE, we take the derivative of the log-likelihood with respect to lambda, set it equal to zero, and solve for lambda.So, let's compute the derivative:d/d lambda [ log L(lambda) ] = sum_{i=1}^N [ (M_i / lambda) - 1 ]Set this equal to zero:sum_{i=1}^N [ (M_i / lambda) - 1 ] = 0Let's distribute the sum:sum_{i=1}^N (M_i / lambda) - sum_{i=1}^N 1 = 0Which simplifies to:(1 / lambda) * sum_{i=1}^N M_i - N = 0Now, solve for lambda:(1 / lambda) * sum_{i=1}^N M_i = NMultiply both sides by lambda:sum_{i=1}^N M_i = N * lambdaThen, divide both sides by N:lambda = (sum_{i=1}^N M_i) / NSo, the MLE of lambda is the average of the observed counts M_i.Wait, that makes sense because in a Poisson distribution, the mean is equal to lambda. So, the MLE is just the sample mean of the counts.Okay, that seems straightforward. So, for the first part, the MLE of lambda is the average of all M_i.Moving on to the second part: We need to estimate the computational time required for a motif search algorithm. The time complexity is given as O(N * L * k). The parameters are N = 10^6, L = 1000, k = 10, and each basic operation takes 10^-9 seconds.So, first, let's compute the total number of operations. Since the time complexity is O(N * L * k), the number of operations is approximately N * L * k.Plugging in the numbers:N = 10^6L = 1000k = 10So, operations = 10^6 * 1000 * 10Let me compute that:10^6 * 1000 = 10^910^9 * 10 = 10^10So, 10^10 operations.Each operation takes 10^-9 seconds. So, total time is operations * time per operation.Total time = 10^10 * 10^-9 = 10^(10 - 9) = 10^1 = 10 seconds.Wait, that seems surprisingly fast. Let me double-check.10^6 sequences, each of length 1000, looking for a motif of length 10. So, for each sequence, the number of possible starting positions is L - k + 1 = 1000 - 10 + 1 = 991. So, per sequence, it's 991 operations. Then, for N sequences, it's N * (L - k + 1) operations. But the time complexity is given as O(N * L * k). Hmm, perhaps the algorithm is more complex than just sliding a window.Wait, perhaps the time complexity is O(N * L * k) because for each position in the sequence, it's comparing k characters. So, for each of the N sequences, for each of the L positions, it's doing k operations. So, that would be N * L * k operations.So, in that case, it's indeed 10^6 * 1000 * 10 = 10^10 operations.Each operation is 10^-9 seconds, so total time is 10^10 * 10^-9 = 10 seconds.That seems correct. So, the total computational time required is 10 seconds.But wait, 10 seconds seems very fast for processing a million sequences. Let me think about it again.Each sequence is 1000 bases long, and for each base, we're checking 10 bases ahead for the motif. So, per sequence, it's 1000 * 10 operations, which is 10,000 operations. For a million sequences, that's 10^6 * 10^4 = 10^10 operations. At 10^-9 seconds per operation, that's 10 seconds. Yeah, that adds up.Alternatively, if we think in terms of FLOPS (floating-point operations per second), but here it's just basic operations, so the same logic applies.So, I think my calculation is correct.Final Answer1. The MLE of ( lambda ) is ( boxed{bar{M}} ), where ( bar{M} ) is the average of the observed motif counts.2. The estimated total computational time is ( boxed{10} ) seconds.</think>

🧠:<think>Okay, so I have this problem about a resistance band designed to enhance vertical jump performance. The entrepreneur and the former jumper are working together to model the force exerted by the band as a function of its elongation. They want to optimize the design for maximum vertical jump height. There are two parts to this problem.Starting with part 1: They mention that the resistance band follows Hooke's Law up to a certain elongation limit ( x_0 ). So, for ( x leq x_0 ), the force is ( F(x) = kx ), where ( k ) is the spring constant. Beyond ( x_0 ), the band exhibits nonlinear behavior, given by ( F(x) = kx_0 + alpha (x - x_0)^2 ), where ( alpha ) is another constant, and ( x > x_0 ). The task is to determine the values of ( x ) that maximize the potential energy stored in the band for ( 0 < x leq L ), where ( L ) is the maximum stretch length.Alright, so potential energy stored in a spring is generally given by the integral of force with respect to displacement. So, for a spring following Hooke's Law, it's ( frac{1}{2}kx^2 ). But here, the force changes beyond ( x_0 ), so the potential energy will be different in two regions: up to ( x_0 ) and beyond ( x_0 ).First, let's write the potential energy function ( U(x) ) as the integral of ( F(x) ) from 0 to ( x ). So, for ( x leq x_0 ), ( U(x) = int_0^x kx' dx' = frac{1}{2}kx^2 ).For ( x > x_0 ), the force is ( F(x) = kx_0 + alpha(x - x_0)^2 ). So, the potential energy will be the integral from 0 to ( x_0 ) plus the integral from ( x_0 ) to ( x ). So, ( U(x) = frac{1}{2}kx_0^2 + int_{x_0}^x [kx_0 + alpha(x' - x_0)^2] dx' ).Let me compute that integral. Let's make a substitution: let ( y = x' - x_0 ), so when ( x' = x_0 ), ( y = 0 ), and when ( x' = x ), ( y = x - x_0 ). So, the integral becomes ( int_0^{x - x_0} [kx_0 + alpha y^2] dy ).Breaking that into two integrals: ( int_0^{x - x_0} kx_0 dy + int_0^{x - x_0} alpha y^2 dy ).The first integral is ( kx_0 (x - x_0) ).The second integral is ( alpha left[ frac{y^3}{3} right]_0^{x - x_0} = alpha frac{(x - x_0)^3}{3} ).So, combining everything, the potential energy for ( x > x_0 ) is:( U(x) = frac{1}{2}kx_0^2 + kx_0(x - x_0) + frac{alpha}{3}(x - x_0)^3 ).Simplify that:First term: ( frac{1}{2}kx_0^2 ).Second term: ( kx_0 x - kx_0^2 ).Third term: ( frac{alpha}{3}(x - x_0)^3 ).So, combining the first and second terms:( frac{1}{2}kx_0^2 - kx_0^2 + kx_0 x = -frac{1}{2}kx_0^2 + kx_0 x ).Therefore, the potential energy is:( U(x) = -frac{1}{2}kx_0^2 + kx_0 x + frac{alpha}{3}(x - x_0)^3 ).So, now, we have the potential energy function defined piecewise:- For ( x leq x_0 ): ( U(x) = frac{1}{2}kx^2 ).- For ( x > x_0 ): ( U(x) = -frac{1}{2}kx_0^2 + kx_0 x + frac{alpha}{3}(x - x_0)^3 ).We need to find the value of ( x ) that maximizes ( U(x) ) for ( 0 < x leq L ).Since ( U(x) ) is a continuous function, we can analyze its behavior in both regions.First, let's consider the region ( x leq x_0 ). The potential energy is ( frac{1}{2}kx^2 ), which is a parabola opening upwards. Its derivative is ( kx ), which is positive for ( x > 0 ). So, in this region, ( U(x) ) is increasing as ( x ) increases. Therefore, the maximum in this region would be at ( x = x_0 ).Now, moving to the region ( x > x_0 ). Here, the potential energy is given by the cubic function above. To find its maximum, we can take the derivative and set it equal to zero.Compute ( dU/dx ) for ( x > x_0 ):( dU/dx = kx_0 + alpha (x - x_0)^2 ).Wait, let me check that derivative. The potential energy is ( U(x) = -frac{1}{2}kx_0^2 + kx_0 x + frac{alpha}{3}(x - x_0)^3 ).So, derivative:( dU/dx = kx_0 + alpha (x - x_0)^2 times 1 ) (since derivative of ( (x - x_0)^3 ) is ( 3(x - x_0)^2 ), so when multiplied by ( frac{alpha}{3} ), it becomes ( alpha (x - x_0)^2 )).So, ( dU/dx = kx_0 + alpha (x - x_0)^2 ).We set this equal to zero to find critical points:( kx_0 + alpha (x - x_0)^2 = 0 ).But ( kx_0 ) is positive because ( k > 0 ) and ( x_0 > 0 ). Also, ( alpha > 0 ), so ( alpha (x - x_0)^2 ) is always non-negative. Therefore, the sum of two positive terms can't be zero. So, there are no critical points in ( x > x_0 ) where the derivative is zero.This suggests that ( U(x) ) is increasing for all ( x > x_0 ) because the derivative is always positive. Therefore, the maximum potential energy in the region ( x > x_0 ) occurs at the maximum stretch ( x = L ).So, comparing the potential energy at ( x = x_0 ) and at ( x = L ), we can determine where the maximum occurs.Compute ( U(x_0) ):From the first region: ( U(x_0) = frac{1}{2}kx_0^2 ).From the second region, plugging ( x = x_0 ):( U(x_0) = -frac{1}{2}kx_0^2 + kx_0 x_0 + frac{alpha}{3}(0)^3 = -frac{1}{2}kx_0^2 + kx_0^2 = frac{1}{2}kx_0^2 ). So, consistent.Now, compute ( U(L) ):( U(L) = -frac{1}{2}kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 ).We need to compare ( U(L) ) and ( U(x_0) ) to see which is larger.So, compute ( U(L) - U(x_0) ):( [ -frac{1}{2}kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 ] - [ frac{1}{2}kx_0^2 ] )Simplify:( -frac{1}{2}kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 - frac{1}{2}kx_0^2 )Combine like terms:( -kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 )Factor out ( kx_0 ):( kx_0 (L - x_0) + frac{alpha}{3}(L - x_0)^3 )Factor out ( (L - x_0) ):( (L - x_0) [ kx_0 + frac{alpha}{3}(L - x_0)^2 ] )Since ( L > x_0 ), ( (L - x_0) > 0 ). Also, ( kx_0 > 0 ) and ( frac{alpha}{3}(L - x_0)^2 > 0 ). Therefore, the entire expression is positive. So, ( U(L) - U(x_0) > 0 ), which means ( U(L) > U(x_0) ).Therefore, the potential energy is maximized at ( x = L ).Wait, but hold on. The problem says "determine the values of ( x ) that maximize the potential energy stored in the band for ( 0 < x leq L )". So, the maximum occurs at ( x = L ).But let me double-check. Since in the region ( x > x_0 ), the potential energy is increasing, and since ( L ) is the maximum stretch, then yes, the maximum potential energy is at ( x = L ).So, the value of ( x ) that maximizes the potential energy is ( x = L ).Wait, but just to be thorough, is there any point where the potential energy could be higher than at ( L )? Since ( U(x) ) is increasing for ( x > x_0 ), and ( L ) is the upper limit, so no, ( U(L) ) is the maximum.Therefore, the answer to part 1 is ( x = L ).Moving on to part 2: The jumper provides data suggesting that the optimal design should maximize the impulse delivered during a jump. Impulse is the integral of force with respect to time over the duration of the jump.Assuming the resistance band starts from rest and elongates linearly with time until it reaches maximum stretch ( L ) in time ( T ). So, the elongation ( x(t) ) is a linear function of time. Let's model that.If it starts from rest, at ( t = 0 ), ( x(0) = 0 ). At ( t = T ), ( x(T) = L ). So, the elongation as a function of time is ( x(t) = frac{L}{T} t ), for ( 0 leq t leq T ).So, the velocity ( v(t) = dx/dt = frac{L}{T} ).But wait, actually, if it's elongating linearly with time, then ( x(t) = vt ), where ( v = L/T ). So, the velocity is constant? That seems a bit odd because in reality, the velocity might change due to the force, but the problem states it's elongating linearly with time, so we can take ( x(t) = (L/T) t ).Given that, we can express the force as a function of time. Since ( x(t) ) is known, we can substitute into the force equation.But first, we need to express the force as a function of ( x(t) ). Depending on whether ( x(t) ) is less than or equal to ( x_0 ) or greater than ( x_0 ), the force will be different.So, let's find the time ( t_1 ) when ( x(t) = x_0 ). That is, ( x_0 = (L/T) t_1 ), so ( t_1 = (x_0 T)/L ).Therefore, for ( t leq t_1 ), ( x(t) leq x_0 ), so the force is ( F(t) = kx(t) = k(L/T) t ).For ( t > t_1 ), ( x(t) > x_0 ), so the force is ( F(t) = kx_0 + alpha (x(t) - x_0)^2 = kx_0 + alpha ( (L/T) t - x_0 )^2 ).So, the impulse ( J ) is the integral of ( F(t) ) from ( t = 0 ) to ( t = T ). So, we can split the integral into two parts: from 0 to ( t_1 ) and from ( t_1 ) to ( T ).So, ( J = int_0^{t_1} F(t) dt + int_{t_1}^T F(t) dt ).Compute each integral separately.First integral: ( int_0^{t_1} k(L/T) t dt ).This is ( k(L/T) times int_0^{t_1} t dt = k(L/T) times [ frac{t^2}{2} ]_0^{t_1} = k(L/T) times frac{t_1^2}{2} ).Substitute ( t_1 = (x_0 T)/L ):( k(L/T) times frac{(x_0 T / L)^2}{2} = k(L/T) times frac{x_0^2 T^2}{2 L^2} = k times frac{x_0^2 T}{2 L} ).Simplify: ( frac{k x_0^2 T}{2 L} ).Second integral: ( int_{t_1}^T [kx_0 + alpha ( (L/T) t - x_0 )^2 ] dt ).Let me make a substitution to simplify. Let ( u = (L/T) t - x_0 ). Then, ( du/dt = L/T ), so ( dt = (T/L) du ).When ( t = t_1 ), ( u = (L/T)(x_0 T / L) - x_0 = x_0 - x_0 = 0 ).When ( t = T ), ( u = (L/T) T - x_0 = L - x_0 ).So, the integral becomes:( int_{u=0}^{u=L - x_0} [kx_0 + alpha u^2 ] times (T/L) du ).Factor out ( T/L ):( (T/L) int_0^{L - x_0} [kx_0 + alpha u^2 ] du ).Compute the integral:( (T/L) [ kx_0 (L - x_0) + alpha times frac{(L - x_0)^3}{3} ] ).Simplify:( (T/L) [ kx_0 (L - x_0) + frac{alpha}{3}(L - x_0)^3 ] ).So, combining both integrals, the total impulse ( J ) is:( frac{k x_0^2 T}{2 L} + (T/L) [ kx_0 (L - x_0) + frac{alpha}{3}(L - x_0)^3 ] ).Let me factor out ( T/L ):( J = frac{T}{L} left( frac{k x_0^2}{2} + kx_0 (L - x_0) + frac{alpha}{3}(L - x_0)^3 right ) ).Simplify inside the brackets:First term: ( frac{k x_0^2}{2} ).Second term: ( kx_0 L - kx_0^2 ).Third term: ( frac{alpha}{3}(L - x_0)^3 ).Combine the first and second terms:( frac{k x_0^2}{2} - kx_0^2 + kx_0 L = -frac{k x_0^2}{2} + kx_0 L ).So, the expression becomes:( J = frac{T}{L} left( -frac{k x_0^2}{2} + kx_0 L + frac{alpha}{3}(L - x_0)^3 right ) ).Wait, that looks familiar. Isn't that similar to the potential energy at ( x = L )?Yes, earlier in part 1, we had ( U(L) = -frac{1}{2}kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 ). So, ( J = frac{T}{L} U(L) ).Therefore, impulse is proportional to the potential energy at maximum stretch multiplied by ( T/L ).But the problem asks to express the impulse as a function of ( k ), ( alpha ), ( x_0 ), ( L ), and ( T ), which we have done.Now, we need to determine the conditions under which this impulse is maximized given ( k > 0 ), ( alpha > 0 ), and ( x_0 < L ).So, to maximize ( J ), since ( J = frac{T}{L} U(L) ), and ( frac{T}{L} ) is a positive constant (given ( T > 0 ), ( L > 0 )), maximizing ( J ) is equivalent to maximizing ( U(L) ).From part 1, we found that ( U(L) = -frac{1}{2}kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 ).So, to maximize ( U(L) ), we can treat it as a function of ( x_0 ), since ( k ), ( alpha ), ( L ), and ( T ) are parameters, but ( x_0 ) is a design variable that can be adjusted.Wait, actually, in the problem statement, are ( k ), ( alpha ), ( x_0 ), ( L ), and ( T ) all variables, or are some fixed? The problem says "determine the conditions under which this impulse is maximized given the constraints ( k > 0 ), ( alpha > 0 ), and ( x_0 < L )."So, it seems that ( k ), ( alpha ), ( x_0 ), ( L ), and ( T ) are all variables, but with ( k > 0 ), ( alpha > 0 ), and ( x_0 < L ). So, we need to find the conditions on these variables to maximize ( J ).But ( J ) is expressed in terms of these variables. So, perhaps we need to find the optimal ( x_0 ) that maximizes ( J ), given ( k ), ( alpha ), ( L ), and ( T ).Wait, but the problem says "determine the conditions under which this impulse is maximized given the constraints ( k > 0 ), ( alpha > 0 ), and ( x_0 < L )." So, it might be that we need to find relationships between ( k ), ( alpha ), ( x_0 ), ( L ), and ( T ) such that ( J ) is maximized.Alternatively, perhaps we can treat ( x_0 ) as a variable and find its optimal value that maximizes ( J ).Let me think. Since ( J ) is proportional to ( U(L) ), and ( U(L) ) is a function of ( x_0 ), we can take the derivative of ( U(L) ) with respect to ( x_0 ) and set it to zero to find the optimal ( x_0 ).So, let's compute ( dU(L)/dx_0 ):( U(L) = -frac{1}{2}kx_0^2 + kx_0 L + frac{alpha}{3}(L - x_0)^3 ).Compute derivative:( dU/dx_0 = -k x_0 + k L + frac{alpha}{3} times 3 (L - x_0)^2 (-1) ).Simplify:( dU/dx_0 = -k x_0 + k L - alpha (L - x_0)^2 ).Set derivative equal to zero for maximization:( -k x_0 + k L - alpha (L - x_0)^2 = 0 ).Let me rearrange:( k(L - x_0) = alpha (L - x_0)^2 ).Assuming ( L neq x_0 ) (since ( x_0 < L )), we can divide both sides by ( (L - x_0) ):( k = alpha (L - x_0) ).Therefore,( L - x_0 = frac{k}{alpha} ).So,( x_0 = L - frac{k}{alpha} ).But we have the constraint ( x_0 < L ), which is satisfied as long as ( frac{k}{alpha} > 0 ), which it is since ( k > 0 ) and ( alpha > 0 ).Therefore, the optimal ( x_0 ) that maximizes the impulse is ( x_0 = L - frac{k}{alpha} ).But we need to ensure that ( x_0 > 0 ), because elongation can't be negative. So,( L - frac{k}{alpha} > 0 implies frac{k}{alpha} < L implies k < alpha L ).So, if ( k < alpha L ), then ( x_0 = L - frac{k}{alpha} ) is positive. If ( k geq alpha L ), then ( x_0 ) would be non-positive, which is not allowed, so in that case, the maximum would occur at ( x_0 = 0 ), but since ( x_0 ) is the point where the nonlinear behavior starts, setting ( x_0 = 0 ) would mean the band is always nonlinear, which might not be practical.But given the constraints ( k > 0 ), ( alpha > 0 ), and ( x_0 < L ), the optimal ( x_0 ) is ( L - frac{k}{alpha} ), provided that ( L - frac{k}{alpha} > 0 ), i.e., ( k < alpha L ).Therefore, the condition for maximum impulse is ( x_0 = L - frac{k}{alpha} ), given ( k < alpha L ).Alternatively, if ( k geq alpha L ), then the maximum impulse occurs at ( x_0 = 0 ), but that might not be a practical design since the band would exhibit nonlinear behavior from the start, which might not be desirable.So, summarizing, to maximize the impulse, the elongation limit ( x_0 ) should be set to ( L - frac{k}{alpha} ), provided that ( k < alpha L ). If ( k geq alpha L ), then ( x_0 ) should be as small as possible, but since ( x_0 ) is a design parameter, it's likely that the optimal design occurs when ( k < alpha L ), allowing ( x_0 ) to be positive.Therefore, the conditions for maximum impulse are ( x_0 = L - frac{k}{alpha} ) with ( k < alpha L ).So, putting it all together:1. The potential energy is maximized at ( x = L ).2. The impulse is maximized when ( x_0 = L - frac{k}{alpha} ) provided ( k < alpha L ).Final Answer1. The potential energy is maximized at ( boxed{L} ).2. The impulse is maximized when ( x_0 = boxed{L - dfrac{k}{alpha}} ) given ( k < alpha L ).</think>

🧠:<think>Okay, so I have this problem where a business traveler from New York needs to visit Boston, Chicago, and San Francisco, each exactly once, and then return to New York. The goal is to find the most efficient route in terms of total flying distance and then calculate the fuel cost for that route. First, I need to understand the problem. It sounds like a variation of the Traveling Salesman Problem (TSP), where the traveler has to visit each city once and return to the starting point, minimizing the total distance. Since there are only four cities involved (including New York), it's a manageable number, so maybe I can list all possible routes and calculate their total distances.Let me list the cities: New York (NY), Boston (B), Chicago (C), San Francisco (SF). The traveler starts and ends in NY, visiting B, C, and SF in between. So, the possible sequences of cities are permutations of B, C, SF. Since there are three cities, there are 3! = 6 possible routes.I should list all these permutations and calculate the total distance for each. The distances between the cities are given, so I can use those.Let me note down all the distances:- NY to B: 215 miles- NY to C: 790 miles- NY to SF: 2,565 miles- B to C: 850 miles- B to SF: 2,705 miles- C to SF: 1,845 milesNow, let's list all possible routes starting and ending at NY. Each route will have four segments: NY to first city, first to second, second to third, third back to NY.The permutations of B, C, SF are:1. B -> C -> SF2. B -> SF -> C3. C -> B -> SF4. C -> SF -> B5. SF -> B -> C6. SF -> C -> BFor each permutation, I'll compute the total distance.Starting with Route 1: B -> C -> SFTotal distance = NY to B + B to C + C to SF + SF to NYPlugging in the numbers:215 (NY-B) + 850 (B-C) + 1,845 (C-SF) + 2,565 (SF-NY)Let me add these up step by step:215 + 850 = 1,0651,065 + 1,845 = 2,9102,910 + 2,565 = 5,475 milesOkay, Route 1 totals 5,475 miles.Route 2: B -> SF -> CTotal distance = NY-B + B-SF + SF-C + C-NYCalculating:215 + 2,705 + 1,845 + 790Adding step by step:215 + 2,705 = 2,9202,920 + 1,845 = 4,7654,765 + 790 = 5,555 milesSo Route 2 is 5,555 miles.Route 3: C -> B -> SFTotal distance = NY-C + C-B + B-SF + SF-NYWait, but C to B is the same as B to C, right? So 850 miles.So, 790 (NY-C) + 850 (C-B) + 2,705 (B-SF) + 2,565 (SF-NY)Adding:790 + 850 = 1,6401,640 + 2,705 = 4,3454,345 + 2,565 = 6,910 milesThat's Route 3: 6,910 miles.Route 4: C -> SF -> BTotal distance = NY-C + C-SF + SF-B + B-NYWait, SF-B is the same as B-SF, which is 2,705 miles.So, 790 (NY-C) + 1,845 (C-SF) + 2,705 (SF-B) + 215 (B-NY)Calculating:790 + 1,845 = 2,6352,635 + 2,705 = 5,3405,340 + 215 = 5,555 milesSo Route 4 is 5,555 miles.Route 5: SF -> B -> CTotal distance = NY-SF + SF-B + B-C + C-NYSo, 2,565 (NY-SF) + 2,705 (SF-B) + 850 (B-C) + 790 (C-NY)Adding:2,565 + 2,705 = 5,2705,270 + 850 = 6,1206,120 + 790 = 6,910 milesRoute 5 totals 6,910 miles.Route 6: SF -> C -> BTotal distance = NY-SF + SF-C + C-B + B-NYWhich is 2,565 + 1,845 + 850 + 215Calculating:2,565 + 1,845 = 4,4104,410 + 850 = 5,2605,260 + 215 = 5,475 milesSo Route 6 is 5,475 miles.Now, compiling all the total distances:1. B -> C -> SF: 5,4752. B -> SF -> C: 5,5553. C -> B -> SF: 6,9104. C -> SF -> B: 5,5555. SF -> B -> C: 6,9106. SF -> C -> B: 5,475Looking at these totals, the minimum distance is 5,475 miles, achieved by Routes 1 and 6.So, the optimal routes are:1. NY -> B -> C -> SF -> NY6. NY -> SF -> C -> B -> NYBoth have the same total distance of 5,475 miles.Therefore, the traveler can choose either of these two routes. But let me just double-check the calculations to make sure I didn't make a mistake.For Route 1: 215 + 850 + 1,845 + 2,565215 + 850 = 1,0651,065 + 1,845 = 2,9102,910 + 2,565 = 5,475. Correct.For Route 6: 2,565 + 1,845 + 850 + 2152,565 + 1,845 = 4,4104,410 + 850 = 5,2605,260 + 215 = 5,475. Correct.So, both are indeed 5,475 miles.Therefore, the minimal total flying distance is 5,475 miles.Moving on to Sub-problem 2: Calculating the total fuel cost.Given that the jet consumes 5 miles per gallon, so fuel consumption is 1 gallon per 5 miles. Therefore, fuel needed is total distance divided by 5.Fuel cost is 4.50 per gallon.So, total fuel cost = (Total distance / 5) * 4.50Plugging in the total distance of 5,475 miles:Fuel needed = 5,475 / 5 = 1,095 gallonsTotal cost = 1,095 * 4.50Let me compute that:1,095 * 4 = 4,3801,095 * 0.5 = 547.5Adding together: 4,380 + 547.5 = 4,927.5So, total fuel cost is 4,927.50But let me check the multiplication again:1,095 * 4.50Alternatively, 1,095 * 4.5 = (1,000 + 95) * 4.5 = 1,000*4.5 + 95*4.5 = 4,500 + 427.5 = 4,927.5Yes, that's correct.So, the total fuel cost is 4,927.50But since we usually don't deal with half dollars in fuel costs, maybe we can round it to the nearest cent, which would still be 4,927.50.Alternatively, if required, it could be expressed as 4,927.50.So, summarizing:Sub-problem 1: The optimal routes are NY -> B -> C -> SF -> NY or NY -> SF -> C -> B -> NY, both totaling 5,475 miles.Sub-problem 2: The total fuel cost is 4,927.50.Final AnswerSub-problem 1: The optimal route is either New York → Boston → Chicago → San Francisco → New York or New York → San Francisco → Chicago → Boston → New York, with a total distance of boxed{5475} miles.Sub-problem 2: The total fuel cost for this route is boxed{4927.50} dollars.</think>