🧠:<think>Okay, so I have this problem about a doctoral student analyzing John Milton's influence on modern literature using a network graph. The graph is represented by an adjacency matrix, and each edge has a weight indicating the strength of influence. The student is looking at eigenvector centrality and a spanning tree subgraph. Hmm, let me try to break this down.Starting with part 1: They want the eigenvector centrality of the network. Eigenvector centrality is a measure of the influence of a node in a network. It's based on the idea that connections to high-scoring nodes contribute more to the score of the node in question. So, in this context, each literary work (node) is influenced by others, and the eigenvector centrality would tell us which works are most influential.The adjacency matrix A has elements a_ij, which are the weights from node i to node j. Since the graph is strongly connected, there's a path from any node to any other node, which is important because it ensures that the eigenvector centrality is well-defined and that there's a unique dominant eigenvector.To find the eigenvector centrality, I remember that it's the eigenvector corresponding to the largest eigenvalue of the adjacency matrix. So, mathematically, we need to solve the equation A * x = λ * x, where x is the eigenvector and λ is the eigenvalue. The eigenvector with the largest λ gives the centrality scores for each node.In the context of Milton's influence, a higher eigenvector centrality for a node (work) means that it's influenced by many other influential works. So, if a modern literary work has a high eigenvector centrality, it suggests that it's significantly influenced by other works that are themselves influential. This could indicate that Milton's works, if they have high centrality, are foundational and have a wide-reaching impact on modern literature.Moving on to part 2: The student found a subgraph which is a spanning tree of a specific literary movement influenced by Milton. A spanning tree is a subgraph that includes all the nodes and is a tree (no cycles). It's rooted at a work by Milton, and the sum of the weights is maximized. So, this is like finding the maximum spanning tree (MST) in the subgraph.Wait, but the subgraph is already a spanning tree, so maybe it's just a specific spanning tree. The total influence would be the sum of the weights of all edges in this spanning tree. Since it's rooted at Milton's work, the edges represent the influence flowing from Milton to other works in the movement.To maximize the total influence, we need to maximize the sum of the edge weights. That would mean selecting the edges with the highest weights that connect all the nodes without forming cycles. So, this is essentially finding the maximum spanning tree of the subgraph, which can be done using algorithms like Krusky's or Prim's.In terms of assessing Milton's significance, a higher total influence (sum of weights) indicates that the literary movement is heavily influenced by Milton's work. The maximum spanning tree ensures that we're considering the strongest influences, so if the total is high, it suggests that Milton's work is central and has a strong, direct influence on the movement. This could be used to argue that Milton's works are foundational or highly influential within that specific literary context.I should also consider if there are any potential issues or assumptions here. For eigenvector centrality, we assume that the adjacency matrix is such that the dominant eigenvalue is positive and unique, which it should be for a strongly connected graph. For the spanning tree, we're assuming that the subgraph is connected, which it is since it's a spanning tree. Also, the weights are between 0 and 1, so maximizing the sum would give a value up to m-1 (since a tree has m-1 edges).Wait, but in the problem, the subgraph is a spanning tree, so it's already connected, and we're just summing the weights. So, the total influence is simply the sum of all the edge weights in this tree. Since it's rooted at Milton, the edges are directed away from Milton, showing the flow of influence.So, putting it all together, eigenvector centrality gives a measure of each work's influence considering the network structure, while the maximum spanning tree gives the total influence of a specific movement, emphasizing the strongest connections from Milton.I think I've got a handle on this. Let me try to write the answers formally now.Problem 1: Eigenvector CentralityThe eigenvector centrality of a node in a network is determined by the eigenvector corresponding to the largest eigenvalue of the adjacency matrix. For a strongly connected graph with adjacency matrix ( A ), the eigenvector centrality vector ( x ) satisfies:[A x = lambda x]where ( lambda ) is the largest eigenvalue. The components of ( x ) represent the centrality scores of each node. In the context of Milton's influence, a higher eigenvector centrality for a literary work indicates that it is influenced by other works that are themselves highly influential, suggesting a significant role in the network of literary influence.Problem 2: Total Influence of a Literary MovementGiven a spanning tree subgraph rooted at Milton's work with ( m ) nodes, the total influence is the sum of the weights of all edges in the spanning tree. To maximize this total influence, we consider the maximum spanning tree (MST) of the subgraph. The total influence ( T ) can be expressed as:[T = sum_{(i,j) in E} a_{ij}]where ( E ) is the set of edges in the maximum spanning tree. This sum represents the strongest influences within the literary movement, with higher values indicating a more significant impact of Milton's work on the movement.Final Answer1. The eigenvector centrality is determined by the dominant eigenvector of the adjacency matrix ( A ). In the context of Milton's influence, it highlights works most influenced by other influential texts. The eigenvector centrality is given by solving ( A x = lambda x ) for the dominant eigenvalue ( lambda ) and corresponding eigenvector ( x ).2. The total influence of the literary movement is the sum of the edge weights in the maximum spanning tree rooted at Milton's work. This is expressed as ( sum_{(i,j) in E} a_{ij} ), where ( E ) is the edge set of the MST. A higher total influence indicates greater significance of Milton's work in the movement.boxed{1. text{Eigenvector centrality is determined by the dominant eigenvector of } A.}boxed{2. text{Total influence is } sum_{(i,j) in E} a_{ij} text{ for the maximum spanning tree.}}</think>

🧠:<think>Alright, so I have this problem about a human resources manager analyzing a job search guidance program. There are two parts to the problem. Let me try to figure them out step by step.First, let me parse the information given. The manager tracks two metrics for each mentee: time to secure a job, denoted as ( T ), and the number of job applications submitted, denoted as ( A ). They collected data from 50 mentees. For the time to secure a job, ( T ), it follows a normal distribution with a mean of 4 months and a standard deviation of 1.5 months. So, ( T sim N(4, 1.5^2) ). For the number of job applications, ( A ), it follows a Poisson distribution with a mean of 20 applications. So, ( A sim text{Poisson}(20) ).Now, moving on to the questions.Question 1: Calculate the probability that a randomly selected mentee secures a job in less than 3 months and submits more than 25 job applications.Hmm, okay. So, we need to find ( P(T < 3 text{ and } A > 25) ). Since ( T ) and ( A ) are two different metrics, I assume they are independent variables. If they are independent, then the joint probability is the product of the individual probabilities. So, ( P(T < 3 text{ and } A > 25) = P(T < 3) times P(A > 25) ).Let me verify if they are independent. The problem doesn't specify any relationship between ( T ) and ( A ), so it's reasonable to assume independence unless stated otherwise.So, first, I need to calculate ( P(T < 3) ). Since ( T ) is normally distributed, I can standardize it and use the standard normal distribution table or a calculator.The formula for standardizing is ( Z = frac{T - mu}{sigma} ). So, plugging in the values:( Z = frac{3 - 4}{1.5} = frac{-1}{1.5} = -0.6667 ).So, ( P(T < 3) = P(Z < -0.6667) ). Looking up this Z-score in the standard normal table, or using a calculator, I can find the probability.I remember that ( P(Z < -0.67) ) is approximately 0.2514. Let me check with a calculator for more precision. Using the Z-table or a calculator, the exact value for Z = -0.6667 is approximately 0.2525. So, roughly 25.25%.Next, I need to calculate ( P(A > 25) ). Since ( A ) follows a Poisson distribution with ( lambda = 20 ), the probability mass function is:( P(A = k) = frac{e^{-lambda} lambda^k}{k!} ).But since we need ( P(A > 25) ), it's easier to calculate ( 1 - P(A leq 25) ). However, calculating this directly might be tedious because it involves summing up probabilities from ( k = 0 ) to ( k = 25 ). Alternatively, since ( lambda = 20 ) is reasonably large, we might approximate the Poisson distribution with a normal distribution. The Poisson distribution can be approximated by ( N(lambda, sqrt{lambda}) ). So, ( A ) can be approximated as ( N(20, sqrt{20}) ). The standard deviation is ( sqrt{20} approx 4.4721 ).So, let's standardize ( A ) for 25.5 (using continuity correction since we're approximating a discrete distribution with a continuous one). So, ( P(A > 25) approx P(A geq 25.5) ).Calculating the Z-score:( Z = frac{25.5 - 20}{4.4721} = frac{5.5}{4.4721} approx 1.2297 ).So, ( P(A geq 25.5) = P(Z geq 1.2297) ). Looking up the Z-table, ( P(Z < 1.23) ) is approximately 0.8907, so ( P(Z geq 1.23) = 1 - 0.8907 = 0.1093 ). So, approximately 10.93%.Therefore, the joint probability is ( 0.2525 times 0.1093 approx 0.0276 ). So, about 2.76%.Wait, but I should verify if the approximation is valid. Since ( lambda = 20 ) is moderately large, the normal approximation should be reasonable, but let me see if I can compute the exact probability.Calculating ( P(A > 25) ) exactly would involve summing ( P(A = k) ) from ( k = 26 ) to infinity. But that's a lot of terms. Alternatively, using the complement, ( 1 - P(A leq 25) ).I can use the cumulative distribution function for Poisson. Maybe I can use a calculator or a table, but since I don't have one, I can use the approximation or perhaps use the normal approximation with continuity correction as before.Alternatively, I can use the fact that for Poisson, the mean and variance are both 20, so the standard deviation is about 4.4721. So, 25 is 5 units above the mean, which is about 1.12 standard deviations away.Wait, but 25 is 5 above the mean of 20, so in terms of standard deviations, it's ( (25 - 20)/4.4721 approx 1.12 ). So, the Z-score is approximately 1.12.Looking up ( P(Z > 1.12) ) is approximately 0.1314. So, about 13.14%.Wait, but earlier with continuity correction, I got 10.93%. So, which one is more accurate?I think the continuity correction is better because we're approximating a discrete distribution with a continuous one. So, 10.93% is more accurate.But let me see, if I use the exact Poisson calculation, what would it be?Calculating ( P(A > 25) = 1 - P(A leq 25) ).Calculating ( P(A leq 25) ) exactly would require summing from 0 to 25. Since that's tedious, maybe I can use an online calculator or a statistical software, but since I don't have access, I can estimate.Alternatively, I can use the normal approximation without continuity correction, which gives a Z-score of ( (25 - 20)/4.4721 approx 1.12 ), so ( P(Z > 1.12) approx 0.1314 ).But with continuity correction, it's 25.5, so Z = 1.2297, which is about 0.1093.So, which one is better? The continuity correction is supposed to give a better approximation, so I think 10.93% is better.Alternatively, perhaps using the Poisson cumulative distribution function in R or Python would give the exact value, but since I can't do that right now, I'll stick with the continuity correction.So, approximately 10.93%.Therefore, the joint probability is approximately 0.2525 * 0.1093 ≈ 0.0276, or 2.76%.Wait, but let me check if that makes sense. If 25% of mentees take less than 3 months, and about 11% submit more than 25 applications, then the chance both happen is about 2.76%. That seems plausible.Alternatively, if I use the exact Poisson probability, it might be slightly different, but without exact computation, I think 10.93% is a reasonable approximation.So, I think the answer is approximately 2.76%.Question 2: If the manager wants to improve the program such that at least 80% of the mentees secure a job in less than ( k ) months, determine the value of ( k ).So, we need to find ( k ) such that ( P(T < k) geq 0.80 ).Since ( T ) is normally distributed with mean 4 and standard deviation 1.5, we can use the inverse normal distribution to find the value of ( k ) such that the cumulative probability is 0.80.In other words, we need to find the 80th percentile of the normal distribution ( N(4, 1.5^2) ).To find this, we can use the inverse Z-table or a calculator.First, find the Z-score corresponding to 0.80 cumulative probability. From the standard normal table, the Z-score for 0.80 is approximately 0.84. Let me verify: ( P(Z < 0.84) ) is approximately 0.7995, which is very close to 0.80. So, Z ≈ 0.84.Then, using the formula:( Z = frac{k - mu}{sigma} )Plugging in the values:( 0.84 = frac{k - 4}{1.5} )Solving for ( k ):( k = 4 + 0.84 * 1.5 )Calculating:0.84 * 1.5 = 1.26So, ( k = 4 + 1.26 = 5.26 ) months.Therefore, the manager needs to set ( k ) to approximately 5.26 months to ensure that at least 80% of mentees secure a job in less than ( k ) months.Wait, but let me double-check the Z-score for 0.80. Using a more precise table, the Z-score for 0.80 is actually approximately 0.8416, which would make ( k = 4 + 0.8416 * 1.5 ).Calculating:0.8416 * 1.5 = 1.2624So, ( k = 4 + 1.2624 = 5.2624 ) months, which is approximately 5.26 months.So, rounding to two decimal places, 5.26 months. Alternatively, if we need a whole number, it would be 5.26, but since the question doesn't specify, we can leave it as is.Alternatively, if we use a calculator for the inverse normal, it might give a more precise value, but 5.26 is a reasonable approximation.So, summarizing:1. The probability is approximately 2.76%.2. The value of ( k ) is approximately 5.26 months.But let me just make sure I didn't make any calculation errors.For question 1:- ( P(T < 3) ): Z = (3 - 4)/1.5 = -0.6667, which corresponds to about 0.2525.- ( P(A > 25) ): Using normal approximation with continuity correction, Z ≈ 1.2297, so probability ≈ 0.1093.Multiplying them: 0.2525 * 0.1093 ≈ 0.0276, which is 2.76%.For question 2:- Finding k such that ( P(T < k) = 0.80 ).- Z ≈ 0.8416, so k = 4 + 0.8416*1.5 ≈ 5.2624.Yes, that seems correct.I think that's it.</think>

🧠:<think>Okay, so I have this problem where Sarah wants to boost her firm's reputation by increasing social media engagement (SME) and enhancing media coverage (MC). The reputation boost R is given by the function R(x, y) = ln(1 + x²) + sqrt(y + 1). She has a constraint 3x + 2y ≤ 30, and both x and y have to be non-negative. I need to find the values of x and y that maximize R under this constraint. Then, I also have to compute the Lagrange multiplier and interpret it.Alright, let's start by understanding the problem. We have an optimization problem with two variables, x and y, and a linear constraint. The function R(x, y) is the reputation boost, which we want to maximize. The constraint is 3x + 2y ≤ 30, which represents the maximum effort Sarah can allocate. So, this is a constrained optimization problem, and I think I can use the method of Lagrange multipliers to solve it.First, let me write down the function and the constraint:Objective function: R(x, y) = ln(1 + x²) + sqrt(y + 1)Constraint: 3x + 2y ≤ 30, with x ≥ 0 and y ≥ 0.Since this is a maximization problem with an inequality constraint, I should check if the maximum occurs at the boundary or within the feasible region. For that, I can use the method of Lagrange multipliers because the maximum is likely to occur at the boundary where 3x + 2y = 30.So, I'll set up the Lagrangian function:L(x, y, λ) = ln(1 + x²) + sqrt(y + 1) - λ(3x + 2y - 30)Wait, actually, the standard form is to subtract λ times the constraint, but since the constraint is 3x + 2y ≤ 30, the Lagrangian should be:L(x, y, λ) = ln(1 + x²) + sqrt(y + 1) + λ(30 - 3x - 2y)But I think it's more common to write it as L = R - λ(3x + 2y - 30). So, maybe I should write it as:L(x, y, λ) = ln(1 + x²) + sqrt(y + 1) - λ(3x + 2y - 30)Either way, the partial derivatives will be similar. Let me proceed with this version.Now, to find the critical points, I need to take the partial derivatives of L with respect to x, y, and λ, and set them equal to zero.First, partial derivative with respect to x:∂L/∂x = (2x)/(1 + x²) - 3λ = 0Similarly, partial derivative with respect to y:∂L/∂y = (1)/(2*sqrt(y + 1)) - 2λ = 0And partial derivative with respect to λ:∂L/∂λ = -(3x + 2y - 30) = 0So, the three equations are:1. (2x)/(1 + x²) - 3λ = 0 --> (2x)/(1 + x²) = 3λ2. 1/(2*sqrt(y + 1)) - 2λ = 0 --> 1/(2*sqrt(y + 1)) = 2λ3. 3x + 2y = 30Now, I need to solve these three equations for x, y, and λ.Let me express λ from the first two equations and set them equal.From equation 1: λ = (2x)/(3(1 + x²))From equation 2: λ = 1/(4*sqrt(y + 1))Therefore:(2x)/(3(1 + x²)) = 1/(4*sqrt(y + 1))Let me write this as:(2x)/(3(1 + x²)) = 1/(4*sqrt(y + 1))Cross-multiplying:8x*sqrt(y + 1) = 3(1 + x²)Hmm, this seems a bit complicated. Maybe I can express y from equation 3 in terms of x and substitute it into this equation.From equation 3: 3x + 2y = 30 --> 2y = 30 - 3x --> y = (30 - 3x)/2So, y = 15 - (3/2)xTherefore, sqrt(y + 1) = sqrt(15 - (3/2)x + 1) = sqrt(16 - (3/2)x)So, sqrt(y + 1) = sqrt(16 - 1.5x)Therefore, plugging back into the equation:8x*sqrt(16 - 1.5x) = 3(1 + x²)This looks like a nonlinear equation in x. Let me denote t = x for simplicity.So, 8t*sqrt(16 - 1.5t) = 3(1 + t²)This seems a bit messy, but maybe I can square both sides to eliminate the square root. However, I have to be careful because squaring can introduce extraneous solutions.Let me write:[8t*sqrt(16 - 1.5t)]² = [3(1 + t²)]²So, 64t²*(16 - 1.5t) = 9(1 + 2t² + t⁴)Let me compute each side:Left side: 64t²*(16 - 1.5t) = 64t²*16 - 64t²*1.5t = 1024t² - 96t³Right side: 9(1 + 2t² + t⁴) = 9 + 18t² + 9t⁴So, bringing all terms to one side:1024t² - 96t³ - 9 - 18t² - 9t⁴ = 0Simplify:-9t⁴ -96t³ + (1024t² - 18t²) -9 = 0Which is:-9t⁴ -96t³ + 1006t² -9 = 0Multiply both sides by -1 to make the leading coefficient positive:9t⁴ + 96t³ - 1006t² +9 = 0So, we have a quartic equation:9t⁴ + 96t³ - 1006t² +9 = 0This seems quite complicated. Maybe I made a mistake earlier? Let me double-check the steps.Starting from the partial derivatives:∂L/∂x = (2x)/(1 + x²) - 3λ = 0 --> (2x)/(1 + x²) = 3λ∂L/∂y = 1/(2*sqrt(y + 1)) - 2λ = 0 --> 1/(2*sqrt(y + 1)) = 2λSo, λ = (2x)/(3(1 + x²)) and λ = 1/(4*sqrt(y + 1))Setting equal:(2x)/(3(1 + x²)) = 1/(4*sqrt(y + 1))Cross multiplying:8x*sqrt(y + 1) = 3(1 + x²)Then, from the constraint, y = 15 - (3/2)x, so sqrt(y + 1) = sqrt(16 - 1.5x)So, 8x*sqrt(16 - 1.5x) = 3(1 + x²)Yes, that seems correct.Then, squaring both sides:64x²*(16 - 1.5x) = 9(1 + 2x² + x⁴)Which gives:1024x² - 96x³ = 9 + 18x² + 9x⁴Bringing all terms to left:-9x⁴ -96x³ + 1006x² -9 = 0Which is the same as 9x⁴ + 96x³ - 1006x² +9 = 0So, quartic equation. Hmm, maybe I can factor this or find rational roots.Using Rational Root Theorem: possible roots are factors of 9 over factors of 9, so ±1, ±3, ±9, ±1/3, etc.Let me test x=1:9(1)^4 +96(1)^3 -1006(1)^2 +9 = 9 +96 -1006 +9 = (9+96+9) -1006 = 114 -1006 = -892 ≠0x=3:9*81 +96*27 -1006*9 +9 = 729 + 2592 -9054 +9 = (729+2592+9) -9054 = 3330 -9054 = -5724 ≠0x=1/3:9*(1/81) +96*(1/27) -1006*(1/9) +9 ≈ 0.111 + 3.555 -111.777 +9 ≈ (0.111 +3.555 +9) -111.777 ≈12.666 -111.777≈-99.111≠0x= -1:9*(-1)^4 +96*(-1)^3 -1006*(-1)^2 +9 =9 -96 -1006 +9= (9+9) - (96+1006)=18 -1102= -1084≠0x= -3:9*81 +96*(-27) -1006*9 +9=729 -2592 -9054 +9= (729+9) - (2592+9054)=738 -11646= -10908≠0x= 1/9:9*(1/6561) +96*(1/729) -1006*(1/81) +9≈0.00135 +0.1316 -12.4198 +9≈ (0.00135 +0.1316 +9) -12.4198≈9.13295 -12.4198≈-3.28685≠0Hmm, none of these seem to be roots. Maybe this quartic doesn't have rational roots. So, perhaps I need to use numerical methods or approximate the solution.Alternatively, maybe I can make a substitution to reduce the quartic to a quadratic in terms of z = x² or something else.Let me see:9x⁴ +96x³ -1006x² +9 =0This is a quartic, but perhaps I can factor it as (ax² + bx + c)(dx² + ex + f). Let me try to factor it.Assume it factors into (3x² + px + q)(3x² + rx + s) = 9x⁴ + (3r + 3p)x³ + (pr + 3s + 3q)x² + (ps + qr)x + qsSet equal to 9x⁴ +96x³ -1006x² +9So, matching coefficients:1. 3r + 3p =96 --> r + p=322. pr + 3s + 3q = -10063. ps + qr =0 (since the coefficient of x is 0)4. qs=9So, from equation 4: qs=9. So, possible integer pairs (q,s) are (1,9),(3,3),(9,1),(-1,-9),(-3,-3),(-9,-1)Let me try q=3, s=3:Then equation 3: p*3 + r*3 =0 --> 3(p + r)=0 --> p + r=0But from equation 1: p + r=32. So, 32=0? Not possible.Next, try q=9, s=1:Equation 3: p*1 + r*9=0 --> p +9r=0Equation 1: p + r=32So, from equation 1: p=32 - rPlug into equation 3: 32 - r +9r=0 -->32 +8r=0 -->8r= -32 -->r= -4Then p=32 - (-4)=36Now, check equation 2: pr +3s +3q=36*(-4) +3*1 +3*9= -144 +3 +27= -114 ≠ -1006Not equal. So, discard.Next, try q= -3, s= -3:Equation 3: p*(-3) + r*(-3)=0 --> -3p -3r=0 --> p + r=0Equation 1: p + r=32. So, 0=32? No.Next, q= -1, s= -9:Equation 3: p*(-9) + r*(-1)=0 --> -9p -r=0 -->9p + r=0Equation 1: p + r=32From equation 1: r=32 - pPlug into equation 3:9p + (32 - p)=0 -->8p +32=0 -->8p= -32 -->p= -4Then r=32 - (-4)=36Check equation 2: pr +3s +3q= (-4)(36) +3*(-9) +3*(-1)= -144 -27 -3= -174 ≠ -1006Nope.Next, q= -9, s= -1:Equation 3: p*(-1) + r*(-9)=0 --> -p -9r=0 -->p +9r=0Equation 1: p + r=32From equation 1: p=32 - rPlug into equation 3:32 - r +9r=0 -->32 +8r=0 -->8r= -32 -->r= -4Then p=32 - (-4)=36Check equation 2: pr +3s +3q=36*(-4) +3*(-1) +3*(-9)= -144 -3 -27= -174 ≠ -1006Nope.How about q=1, s=9:Equation 3: p*9 + r*1=0 -->9p + r=0Equation 1: p + r=32From equation 1: r=32 - pPlug into equation 3:9p +32 - p=0 -->8p +32=0 -->8p= -32 -->p= -4Then r=32 - (-4)=36Check equation 2: pr +3s +3q= (-4)(36) +3*9 +3*1= -144 +27 +3= -114 ≠ -1006Nope.Similarly, q= -1, s=9:Equation 3: p*9 + r*(-1)=0 -->9p - r=0Equation 1: p + r=32From equation 3: r=9pPlug into equation 1: p +9p=32 -->10p=32 -->p=3.2Then r=9*3.2=28.8Check equation 2: pr +3s +3q=3.2*28.8 +3*9 +3*(-1)=92.16 +27 -3=116.16 ≠ -1006Nope.This seems not working. Maybe the quartic doesn't factor nicely, so perhaps I need to use numerical methods.Alternatively, maybe I can use substitution or another approach.Wait, perhaps instead of substituting y in terms of x, I can express λ from both equations and set them equal, then express y in terms of x, and substitute back into the constraint.Wait, that's what I did earlier, leading to the quartic equation.Alternatively, maybe I can use substitution in another way.Let me denote u = x², v = y + 1.Then, R = ln(1 + u) + sqrt(v)But not sure if this helps.Alternatively, maybe I can use substitution for the ratio of the partial derivatives.From the first-order conditions:(2x)/(1 + x²) = 3λand1/(2*sqrt(y + 1)) = 2λSo, let me take the ratio of these two equations:[(2x)/(1 + x²)] / [1/(2*sqrt(y + 1))] = (3λ)/(2λ) = 3/2So,(2x)/(1 + x²) * 2*sqrt(y + 1) = 3/2Simplify:(4x*sqrt(y + 1))/(1 + x²) = 3/2Multiply both sides by (1 + x²):4x*sqrt(y + 1) = (3/2)(1 + x²)Which is the same as before, 8x*sqrt(y + 1) = 3(1 + x²)So, same equation.So, I think I have to proceed numerically.Let me define f(x) = 8x*sqrt(16 - 1.5x) - 3(1 + x²)We need to find x such that f(x)=0.Let me compute f(x) for some x values.First, note that x must satisfy 16 -1.5x ≥0 --> x ≤16/1.5≈10.6667Also, x must be ≥0.So, x ∈ [0, 10.6667]Let me try x=2:sqrt(16 -3)=sqrt(13)=≈3.6055f(2)=8*2*3.6055 -3*(1 +4)= 57.688 -15=42.688>0x=4:sqrt(16 -6)=sqrt(10)=≈3.1623f(4)=8*4*3.1623 -3*(1 +16)= 101.1936 -51=50.1936>0x=5:sqrt(16 -7.5)=sqrt(8.5)=≈2.9155f(5)=8*5*2.9155 -3*(1 +25)= 116.62 -78=38.62>0x=6:sqrt(16 -9)=sqrt(7)=≈2.6458f(6)=8*6*2.6458 -3*(1 +36)= 126.9456 -111=15.9456>0x=7:sqrt(16 -10.5)=sqrt(5.5)=≈2.3452f(7)=8*7*2.3452 -3*(1 +49)= 129.9056 -150= -20.0944<0So, f(6)=15.9456>0, f(7)=-20.0944<0. So, the root is between 6 and7.Let me try x=6.5:sqrt(16 -9.75)=sqrt(6.25)=2.5f(6.5)=8*6.5*2.5 -3*(1 +42.25)= 130 -3*43.25=130 -129.75=0.25≈0.25>0x=6.5 gives f(x)=0.25x=6.6:sqrt(16 -9.9)=sqrt(6.1)=≈2.4698f(6.6)=8*6.6*2.4698 -3*(1 +43.56)= 8*6.6*2.4698≈8*16.270≈130.16 -3*44.56≈130.16 -133.68≈-3.52So, f(6.6)≈-3.52So, the root is between 6.5 and6.6At x=6.5, f=0.25At x=6.55:sqrt(16 -1.5*6.55)=sqrt(16 -9.825)=sqrt(6.175)=≈2.485f(6.55)=8*6.55*2.485 -3*(1 + (6.55)^2)Compute 8*6.55=52.4; 52.4*2.485≈52.4*2 +52.4*0.485≈104.8 +25.434≈130.234Compute 3*(1 +42.9025)=3*43.9025≈131.7075So, f(6.55)=130.234 -131.7075≈-1.4735Still negative.x=6.525:sqrt(16 -1.5*6.525)=sqrt(16 -9.7875)=sqrt(6.2125)=≈2.4925f(6.525)=8*6.525*2.4925 -3*(1 + (6.525)^2)Compute 8*6.525=52.2; 52.2*2.4925≈52.2*2 +52.2*0.4925≈104.4 +25.737≈130.137Compute 3*(1 +42.5656)=3*43.5656≈130.6968So, f(6.525)=130.137 -130.6968≈-0.5598Still negative.x=6.51:sqrt(16 -1.5*6.51)=sqrt(16 -9.765)=sqrt(6.235)=≈2.497f(6.51)=8*6.51*2.497 -3*(1 + (6.51)^2)Compute 8*6.51=52.08; 52.08*2.497≈52.08*2 +52.08*0.497≈104.16 +25.86≈129.02Wait, that can't be right because 52.08*2.497 is actually:52.08 *2=104.1652.08*0.497≈52.08*0.5=26.04, minus 52.08*0.003≈0.156, so≈26.04 -0.156≈25.884So, total≈104.16 +25.884≈130.044Compute 3*(1 +42.3801)=3*43.3801≈130.1403So, f(6.51)=130.044 -130.1403≈-0.0963Almost zero, but still slightly negative.x=6.505:sqrt(16 -1.5*6.505)=sqrt(16 -9.7575)=sqrt(6.2425)=≈2.4985f(6.505)=8*6.505*2.4985 -3*(1 + (6.505)^2)Compute 8*6.505=52.04; 52.04*2.4985≈52.04*2 +52.04*0.4985≈104.08 +25.91≈130.0Compute 3*(1 + (6.505)^2)=3*(1 +42.325)=3*43.325≈129.975So, f(6.505)=130.0 -129.975≈0.025So, f(6.505)=≈0.025>0So, between x=6.505 and x=6.51, f(x) crosses zero.Using linear approximation:At x=6.505, f=0.025At x=6.51, f≈-0.0963So, the root is approximately at x=6.505 + (0 -0.025)*(6.51 -6.505)/(-0.0963 -0.025)Compute delta_x=6.51 -6.505=0.005delta_f= -0.0963 -0.025= -0.1213So, the fraction is (0 -0.025)/(-0.1213)=0.025/0.1213≈0.206So, root≈6.505 +0.206*0.005≈6.505 +0.00103≈6.506So, approximately x≈6.506Let me check x=6.506:sqrt(16 -1.5*6.506)=sqrt(16 -9.759)=sqrt(6.241)=≈2.4982f(6.506)=8*6.506*2.4982 -3*(1 + (6.506)^2)Compute 8*6.506=52.048; 52.048*2.4982≈52.048*2 +52.048*0.4982≈104.096 +25.91≈130.006Compute 3*(1 +42.332)=3*43.332≈129.996So, f(6.506)=130.006 -129.996≈0.01Still slightly positive.x=6.507:sqrt(16 -1.5*6.507)=sqrt(16 -9.7605)=sqrt(6.2395)=≈2.4979f(6.507)=8*6.507*2.4979 -3*(1 + (6.507)^2)Compute 8*6.507=52.056; 52.056*2.4979≈52.056*2 +52.056*0.4979≈104.112 +25.89≈130.002Compute 3*(1 +42.341)=3*43.341≈130.023So, f(6.507)=130.002 -130.023≈-0.021So, f(6.507)=≈-0.021So, between x=6.506 and x=6.507, f(x) crosses zero.Using linear approximation:At x=6.506, f=0.01At x=6.507, f=-0.021So, delta_x=0.001, delta_f=-0.031We need to find x where f=0.So, fraction= (0 -0.01)/(-0.031)=0.01/0.031≈0.3226So, root≈6.506 +0.3226*0.001≈6.506 +0.0003226≈6.5063So, approximately x≈6.5063So, x≈6.506Therefore, x≈6.506Then, y=15 - (3/2)x≈15 -1.5*6.506≈15 -9.759≈5.241So, y≈5.241So, x≈6.506, y≈5.241Now, let me check if these values satisfy the original equation.Compute 8x*sqrt(y +1)=8*6.506*sqrt(5.241 +1)=8*6.506*sqrt(6.241)=8*6.506*2.498≈8*6.506*2.498≈8*16.24≈129.92Compute 3(1 +x²)=3*(1 + (6.506)^2)=3*(1 +42.332)=3*43.332≈129.996So, 129.92≈129.996, which is close, considering the approximation.So, these values are approximately correct.Therefore, the optimal x≈6.506, y≈5.241But let me check if these are indeed maxima.We can check the second derivative or use the bordered Hessian, but since it's a constrained optimization, the bordered Hessian is the way to go.But maybe it's easier to consider that since the feasible region is convex and the objective function is concave, the critical point found is indeed a maximum.Wait, is R(x,y) concave?Let me check the Hessian matrix of R.Compute the second partial derivatives.R_xx = derivative of (2x)/(1 +x²) with respect to x:= [2(1 +x²) -2x*2x]/(1 +x²)^2 = [2 +2x² -4x²]/(1 +x²)^2 = (2 -2x²)/(1 +x²)^2Similarly, R_yy = derivative of 1/(2*sqrt(y +1)) with respect to y:= (-1)/(4*(y +1)^(3/2))R_xy = R_yx = 0, since R is additive in x and y.So, the Hessian matrix is:[ (2 -2x²)/(1 +x²)^2 , 0 ][ 0 , (-1)/(4*(y +1)^(3/2)) ]Since both diagonal elements are negative for x≠0 and y≥0, the Hessian is negative definite, so R is concave.Therefore, the critical point found is a global maximum.So, the optimal x≈6.506, y≈5.241But let me compute more accurately.Wait, x≈6.506, so x≈6.506, which is approximately 6.506 thousands, so 6506 units.Similarly, y≈5.241, which is 524.1 units.But let me see if I can express this more precisely.Alternatively, maybe I can use substitution in terms of λ.From equation 1: λ=(2x)/(3(1 +x²))From equation 2: λ=1/(4*sqrt(y +1))So, equate:(2x)/(3(1 +x²))=1/(4*sqrt(y +1))From the constraint, y=(30 -3x)/2So, sqrt(y +1)=sqrt((30 -3x)/2 +1)=sqrt((32 -3x)/2)So, sqrt((32 -3x)/2)=sqrt(16 -1.5x)So, same as before.So, 8x*sqrt(16 -1.5x)=3(1 +x²)This is the same equation.So, perhaps I can use Newton-Raphson method to solve for x.Let me define f(x)=8x*sqrt(16 -1.5x) -3(1 +x²)We need to find x such that f(x)=0.We can use Newton-Raphson:x_{n+1}=x_n - f(x_n)/f’(x_n)Compute f’(x):f’(x)=8*sqrt(16 -1.5x) +8x*( -1.5/(2*sqrt(16 -1.5x)) ) -6xSimplify:=8*sqrt(16 -1.5x) - (6x)/sqrt(16 -1.5x) -6xSo, f’(x)=8*sqrt(16 -1.5x) - (6x)/sqrt(16 -1.5x) -6xLet me compute f(6.506)=≈0.01f’(6.506)=8*sqrt(16 -9.759) - (6*6.506)/sqrt(16 -9.759) -6*6.506Compute sqrt(6.241)=≈2.498So, f’(6.506)=8*2.498 - (39.036)/2.498 -39.036=19.984 -15.62 -39.036≈19.984 -54.656≈-34.672So, f’(6.506)≈-34.672Then, Newton-Raphson update:x_{n+1}=6.506 - (0.01)/(-34.672)=6.506 +0.000288≈6.506288So, x≈6.506288Compute f(6.506288)=8*6.506288*sqrt(16 -1.5*6.506288) -3*(1 + (6.506288)^2)Compute 16 -1.5*6.506288=16 -9.759432=6.240568sqrt(6.240568)=≈2.498113So, 8*6.506288=52.050352.0503*2.498113≈52.0503*2 +52.0503*0.498113≈104.1006 +25.906≈130.0066Compute 3*(1 + (6.506288)^2)=3*(1 +42.332)=3*43.332≈129.996So, f(x)=130.0066 -129.996≈0.0106Wait, but we expected f(x)=0.01 at x=6.506, but after the update, it's still≈0.0106. Maybe I need more iterations.Wait, perhaps I made a mistake in the derivative.Wait, f’(x)=8*sqrt(16 -1.5x) - (6x)/sqrt(16 -1.5x) -6xAt x=6.506288:sqrt(16 -1.5x)=sqrt(6.240568)=≈2.498113So, f’(x)=8*2.498113 - (6*6.506288)/2.498113 -6*6.506288Compute each term:8*2.498113≈19.98496*6.506288≈39.037739.0377/2.498113≈15.626*6.506288≈39.0377So, f’(x)=19.9849 -15.62 -39.0377≈19.9849 -54.6577≈-34.6728So, f’(x)=≈-34.6728So, x_{n+1}=6.506288 - (0.0106)/(-34.6728)=6.506288 +0.000306≈6.506594Compute f(6.506594):sqrt(16 -1.5*6.506594)=sqrt(16 -9.759891)=sqrt(6.240109)=≈2.4980228*6.506594≈52.0527552.05275*2.498022≈52.05275*2 +52.05275*0.498022≈104.1055 +25.905≈130.01053*(1 + (6.506594)^2)=3*(1 +42.333)=3*43.333≈129.999So, f(x)=130.0105 -129.999≈0.0115Hmm, seems like it's oscillating around 0.01. Maybe I need a better initial guess or perhaps the function is flat near the root.Alternatively, maybe I can accept x≈6.506 as the approximate solution.So, x≈6.506, y≈5.241Therefore, the optimal values are approximately x≈6.506 and y≈5.241But let me check if these are indeed the maxima.Alternatively, maybe I can test the endpoints.The feasible region is a polygon with vertices at (0,0), (0,15), (10,0), and the intersection point of 3x +2y=30 with axes.Wait, actually, the constraint is 3x +2y ≤30, so the feasible region is a triangle with vertices at (0,0), (10,0), and (0,15).So, the maximum can occur either at the critical point inside the region or at one of the vertices.We already found the critical point at (6.506,5.241). Let me compute R at this point and at the vertices.Compute R(6.506,5.241)=ln(1 + (6.506)^2) + sqrt(5.241 +1)=ln(1 +42.332)+sqrt(6.241)=ln(43.332)+2.498≈3.769 +2.498≈6.267Compute R(0,0)=ln(1 +0)+sqrt(0 +1)=0 +1=1R(10,0)=ln(1 +100)+sqrt(0 +1)=ln(101)+1≈4.615 +1=5.615R(0,15)=ln(1 +0)+sqrt(15 +1)=0 +sqrt(16)=4So, R at the critical point≈6.267 is higher than at all vertices. Therefore, the maximum occurs at the critical point.Therefore, the optimal values are x≈6.506, y≈5.241But let me express these more accurately.Given that x≈6.506, which is approximately 6.506 thousands, so 6506 units.Similarly, y≈5.241, which is approximately 524.1 units.But since the problem mentions units as x in thousands and y in hundreds, we can write x≈6.506 (thousands) and y≈5.241 (hundreds).But perhaps we can express these as fractions or exact decimals.Alternatively, maybe I can express x as 6.5 and y as 5.25, but let me check.Wait, x=6.5:sqrt(16 -1.5*6.5)=sqrt(16 -9.75)=sqrt(6.25)=2.5So, f(6.5)=8*6.5*2.5 -3*(1 +6.5²)=130 -3*(1 +42.25)=130 -129.75=0.25So, f(6.5)=0.25>0x=6.506:f(x)=≈0.01So, x=6.506 is very close to 6.5, but slightly higher.Similarly, y=5.241 is close to 5.25.So, perhaps we can write x≈6.506 and y≈5.241, but maybe the problem expects an exact form.Alternatively, perhaps I can solve for x and y in terms of λ.From equation 1: λ=(2x)/(3(1 +x²))From equation 2: λ=1/(4*sqrt(y +1))So, equate:(2x)/(3(1 +x²))=1/(4*sqrt(y +1))From the constraint: y=(30 -3x)/2So, sqrt(y +1)=sqrt((30 -3x)/2 +1)=sqrt((32 -3x)/2)So, sqrt((32 -3x)/2)=sqrt(16 -1.5x)So, 8x*sqrt(16 -1.5x)=3(1 +x²)This is the same equation as before.Alternatively, maybe I can express x in terms of y or vice versa.But I think it's not possible to solve this equation analytically, so numerical methods are necessary.Therefore, the optimal values are approximately x≈6.506 and y≈5.241Now, moving to sub-problem 2: Compute the value of the Lagrange multiplier associated with the constraint at the optimal point found in sub-problem 1, and interpret its meaning.From equation 1: λ=(2x)/(3(1 +x²))At x≈6.506, compute λ:λ=(2*6.506)/(3*(1 + (6.506)^2))≈(13.012)/(3*(1 +42.332))≈13.012/(3*43.332)≈13.012/129.996≈0.1001So, λ≈0.1001Alternatively, from equation 2: λ=1/(4*sqrt(y +1))≈1/(4*sqrt(5.241 +1))≈1/(4*2.498)≈1/9.992≈0.1001Same result.So, λ≈0.1001Interpretation: The Lagrange multiplier λ represents the rate at which the maximum reputation boost R increases per unit increase in the constraint 3x +2y. In other words, it is the shadow price of the constraint. Here, λ≈0.1001 means that if Sarah were to increase her total effort (the right-hand side of the constraint) by one unit, the maximum reputation boost would increase by approximately 0.1001 units.Alternatively, it can be interpreted as the marginal gain in reputation per additional unit of effort allocated to the strategies.So, summarizing:1. The optimal values are x≈6.506 (thousands) and y≈5.241 (hundreds).2. The Lagrange multiplier λ≈0.1001, indicating the marginal increase in reputation per unit increase in effort.But let me check if the units are correct.The constraint is 3x +2y ≤30, where x is in thousands and y is in hundreds.So, 3x is in thousands*3, and 2y is in hundreds*2. To make them compatible, perhaps the units are normalized.Wait, actually, the constraint is 3x +2y ≤30, but x is in thousands and y is in hundreds.So, 3x is in thousands*3, and 2y is in hundreds*2. To add them, they need to be in the same units.Wait, perhaps the constraint is in some combined unit, but it's more likely that the coefficients 3 and 2 are just weights, not units.So, the Lagrange multiplier λ is in units of R per unit of the constraint. Since R is unitless (as it's a logarithm and square root), and the constraint is in units of 3x +2y, which is a combination of thousands and hundreds, but since it's a linear combination, the units are mixed.But perhaps it's better to say that λ is the change in R per unit change in the constraint, regardless of units.Therefore, λ≈0.1001 means that if Sarah increases her total effort (the right-hand side of the constraint) by 1 unit, the maximum reputation R would increase by approximately 0.1001.But since the constraint is 3x +2y ≤30, increasing the RHS by 1 would allow for a slight increase in x and/or y, leading to a marginal increase in R.Therefore, the interpretation is that λ≈0.1001 is the rate at which R increases per unit increase in the total effort allocated to the strategies.So, to summarize:1. The optimal values are x≈6.506 (thousands) and y≈5.241 (hundreds).2. The Lagrange multiplier λ≈0.1001, indicating the marginal gain in reputation per additional unit of effort.But let me check if I can express x and y more precisely.Alternatively, maybe I can use the exact values from the equations.From equation 1: λ=(2x)/(3(1 +x²))From equation 2: λ=1/(4*sqrt(y +1))So, equate:(2x)/(3(1 +x²))=1/(4*sqrt(y +1))From the constraint: y=(30 -3x)/2So, sqrt(y +1)=sqrt((30 -3x)/2 +1)=sqrt((32 -3x)/2)So, 8x*sqrt((32 -3x)/2)=3(1 +x²)Which is the same as 8x*sqrt(16 -1.5x)=3(1 +x²)This is the same equation as before.So, I think the approximate values are the best we can do.Therefore, the final answers are:1. x≈6.506 (thousands), y≈5.241 (hundreds)2. λ≈0.1001But let me check if I can express x and y as fractions.Alternatively, maybe I can express x as 6.5 and y as 5.25, but as we saw earlier, x=6.5 gives f(x)=0.25, which is not zero, so it's not the exact solution.Alternatively, maybe I can write x=6.506 and y=5.241 as exact decimals.But perhaps the problem expects an exact form, but given the quartic equation, it's unlikely.Therefore, the optimal values are approximately x≈6.506 and y≈5.241.So, rounding to three decimal places, x≈6.506 and y≈5.241.But let me check if the problem expects integer values or if decimal values are acceptable.The problem says x is in thousands and y is in hundreds, so decimal values are acceptable.Therefore, the answers are:1. x≈6.506, y≈5.2412. λ≈0.1001But let me check if I can write these as fractions.Alternatively, maybe I can write x=6.5 and y=5.25, but as we saw, x=6.5 gives f(x)=0.25, which is not zero, so it's not the exact solution.Alternatively, maybe I can write x=6.506 and y=5.241 as exact decimals.But perhaps the problem expects an exact form, but given the quartic equation, it's unlikely.Therefore, the optimal values are approximately x≈6.506 and y≈5.241.So, rounding to three decimal places, x≈6.506 and y≈5.241.But let me check if the problem expects integer values or if decimal values are acceptable.The problem says x is in thousands and y is in hundreds, so decimal values are acceptable.Therefore, the answers are:1. x≈6.506 (thousands), y≈5.241 (hundreds)2. λ≈0.1001But to be precise, maybe I can write x≈6.506, y≈5.241, and λ≈0.1001Alternatively, maybe I can write x=6.506, y=5.241, and λ=0.1001But let me check if I can express λ as a fraction.From λ≈0.1001, which is approximately 1/10.So, λ≈1/10=0.1But in reality, it's≈0.1001, which is very close to 0.1.So, maybe the problem expects λ=0.1But let me check:From equation 1: λ=(2x)/(3(1 +x²))At x=6.506, compute 2x=13.0121 +x²=1 +42.332=43.332So, λ=13.012/(3*43.332)=13.012/129.996≈0.1001So, yes, it's≈0.1001, which is≈0.1Therefore, the Lagrange multiplier is approximately 0.1.So, summarizing:1. The optimal values are x≈6.506 (thousands) and y≈5.241 (hundreds).2. The Lagrange multiplier λ≈0.1, indicating that each additional unit of effort allocated to the strategies increases the reputation boost by approximately 0.1 units.Therefore, the final answers are:1. x≈6.506, y≈5.2412. λ≈0.1But to be precise, maybe I should keep more decimal places.Alternatively, maybe I can write x=6.506, y=5.241, and λ=0.1001But I think for the purposes of this problem, rounding to three decimal places is sufficient.So, final answers:1. x≈6.506, y≈5.2412. λ≈0.1001But let me check if I can write these as fractions.Alternatively, maybe I can write x=6.506, y=5.241, and λ=0.1001But I think that's acceptable.So, to conclude:The optimal values are approximately x=6.506 (thousands) and y=5.241 (hundreds), and the Lagrange multiplier is approximately 0.1001.Therefore, the answers are:1. x≈6.506, y≈5.2412. λ≈0.1001But let me check if I can express these in a boxed format as per the instructions.So, the final answers are:1. x≈6.506, y≈5.2412. λ≈0.1001But since the problem might expect exact values, but given the complexity, approximate values are acceptable.Therefore, the boxed answers are:1. boxed{x approx 6.506}, boxed{y approx 5.241}2. boxed{lambda approx 0.1001}But perhaps the problem expects the answers in a single box each.Alternatively, maybe I can write them together.But I think the instructions say to put the final answer within boxed{}, so perhaps each sub-problem's answer in a box.But since there are two sub-problems, I think I need to provide two boxed answers.So, for sub-problem 1, the optimal x and y.For sub-problem 2, the Lagrange multiplier.Therefore, the answers are:1. The optimal values are x≈6.506 and y≈5.241, so boxed{x approx 6.506} and boxed{y approx 5.241}2. The Lagrange multiplier is approximately 0.1001, so boxed{lambda approx 0.1001}Alternatively, if the problem expects exact forms, but given the quartic equation, it's unlikely, so approximate decimal values are acceptable.Therefore, the final answers are as above.</think>

🧠:<think>Alright, so I have this problem about modeling heart rate response during high-intensity interval training (HIIT). It's divided into two parts. Let me try to tackle them one by one.Starting with part 1: The differential equation given is dH/dt = -k(H - H_r) + I(t). I know that H(t) is the heart rate, H_r is the resting heart rate, k is a positive constant, and I(t) is the exercise intensity function. They mentioned that I(t) is periodic with period T, where the first half is I0 and the second half is zero. So, it's like a square wave with duty cycle 50%.I need to find the steady-state periodic solution H(t) over one complete cycle. Hmm, steady-state solution for a linear differential equation with periodic input. I remember that for such cases, the solution will also be periodic with the same period as the input. So, if I can find a particular solution that matches the periodicity, that should be the steady-state.Since I(t) is piecewise constant, maybe I can solve the differential equation in two intervals: during exercise (first half) and during rest (second half). Then, ensure continuity at the boundary.Let me denote the period T as 2τ, so each interval (exercise and rest) is τ. So, τ = T/2. That might make the math a bit cleaner.So, for t in [0, τ), I(t) = I0, and for t in [τ, 2τ), I(t) = 0.First, let's solve the differential equation during exercise:dH/dt = -k(H - H_r) + I0This is a linear ODE. Let me rewrite it:dH/dt + kH = kH_r + I0The integrating factor is e^{∫k dt} = e^{kt}Multiply both sides:e^{kt} dH/dt + k e^{kt} H = (k H_r + I0) e^{kt}Left side is d/dt [e^{kt} H]So, integrate both sides:e^{kt} H = ∫(k H_r + I0) e^{kt} dt + CCompute the integral:∫(k H_r + I0) e^{kt} dt = (k H_r + I0) ∫e^{kt} dt = (k H_r + I0) (e^{kt}/k) + CSo,e^{kt} H = (k H_r + I0) (e^{kt}/k) + CDivide both sides by e^{kt}:H(t) = (k H_r + I0)/k + C e^{-kt}Simplify:H(t) = H_r + I0/k + C e^{-kt}Now, this is the general solution during exercise. We need to find the constant C. But since we're looking for the steady-state solution, we need to consider the solution over multiple periods. So, the solution will approach a periodic function as t increases. Therefore, the transient term (C e^{-kt}) should die out, but since we're considering one period, maybe we need to ensure continuity at t=τ.Wait, perhaps another approach is better. Since the input is periodic, the solution will also be periodic. So, we can write H(t) as a sum of a particular solution and the homogeneous solution. But for steady-state, the homogeneous solution (transient) should have decayed, so we can focus on the particular solution.Alternatively, maybe using Laplace transforms? But since it's a periodic function, maybe Fourier series? Hmm, but I think for a piecewise constant input, we can solve it in each interval and match the solutions at the boundaries.So, let's proceed step by step.First, during the exercise phase, t ∈ [0, τ):dH/dt = -k(H - H_r) + I0We can write this as dH/dt + kH = k H_r + I0We can solve this using integrating factor as above.The solution is H(t) = H_r + I0/k + (H(0) - H_r - I0/k) e^{-kt}Similarly, during the rest phase, t ∈ [τ, 2τ):dH/dt = -k(H - H_r) + 0 = -k(H - H_r)So, dH/dt + kH = k H_rAgain, integrating factor e^{kt}:d/dt [e^{kt} H] = k H_r e^{kt}Integrate:e^{kt} H = k H_r ∫e^{kt} dt = k H_r (e^{kt}/k) + C = H_r e^{kt} + CSo, H(t) = H_r + C e^{-kt}Now, we need to find H(τ) from the exercise phase and set it equal to H(τ) from the rest phase.Let me denote H(τ-) as the limit from the left (end of exercise) and H(τ+) as the limit from the right (start of rest). For the solution to be continuous, H(τ-) = H(τ+).So, first, find H(τ-) from the exercise solution:H(τ) = H_r + I0/k + (H(0) - H_r - I0/k) e^{-kτ}But since we're looking for the steady-state solution, the initial condition H(0) should be equal to H(2τ), because after one period, the solution repeats. So, H(0) = H(2τ).Similarly, H(τ+) = H_r + C e^{-kτ}But also, H(τ+) must equal H(τ-). So,H(τ-) = H_r + I0/k + (H(0) - H_r - I0/k) e^{-kτ} = H(τ+) = H_r + C e^{-kτ}So, equate them:H_r + I0/k + (H(0) - H_r - I0/k) e^{-kτ} = H_r + C e^{-kτ}Subtract H_r from both sides:I0/k + (H(0) - H_r - I0/k) e^{-kτ} = C e^{-kτ}Let me solve for C:C e^{-kτ} = I0/k + (H(0) - H_r - I0/k) e^{-kτ}Divide both sides by e^{-kτ}:C = I0/k e^{kτ} + H(0) - H_r - I0/kSo, C = H(0) - H_r + I0/k (e^{kτ} - 1)Now, let's look at the rest phase solution:H(t) = H_r + C e^{-kt} for t ∈ [τ, 2τ)At t=2τ, we have H(2τ) = H_r + C e^{-k(2τ)}But since H(2τ) = H(0) (steady-state), we can write:H(0) = H_r + C e^{-2kτ}Substitute C from above:H(0) = H_r + [H(0) - H_r + I0/k (e^{kτ} - 1)] e^{-2kτ}Let me expand this:H(0) = H_r + H(0) e^{-2kτ} - H_r e^{-2kτ} + I0/k (e^{kτ} - 1) e^{-2kτ}Bring all terms involving H(0) to the left:H(0) - H(0) e^{-2kτ} = H_r - H_r e^{-2kτ} + I0/k (e^{-kτ} - e^{-2kτ})Factor H(0):H(0) [1 - e^{-2kτ}] = H_r [1 - e^{-2kτ}] + I0/k (e^{-kτ} - e^{-2kτ})Divide both sides by [1 - e^{-2kτ}]:H(0) = H_r + I0/k (e^{-kτ} - e^{-2kτ}) / [1 - e^{-2kτ}]Simplify the fraction:(e^{-kτ} - e^{-2kτ}) / (1 - e^{-2kτ}) = e^{-kτ}(1 - e^{-kτ}) / (1 - e^{-2kτ})Note that 1 - e^{-2kτ} = (1 - e^{-kτ})(1 + e^{-kτ})So,= e^{-kτ}(1 - e^{-kτ}) / [(1 - e^{-kτ})(1 + e^{-kτ})] = e^{-kτ} / (1 + e^{-kτ})Therefore,H(0) = H_r + I0/k * e^{-kτ} / (1 + e^{-kτ})Simplify:H(0) = H_r + (I0/k) * e^{-kτ} / (1 + e^{-kτ})Alternatively, factor e^{-kτ}:= H_r + (I0/k) * 1 / (e^{kτ} + 1)Because e^{-kτ}/(1 + e^{-kτ}) = 1/(e^{kτ} + 1)So, H(0) = H_r + (I0/k) / (1 + e^{kτ})Now, with H(0) known, we can write the solutions in both intervals.During exercise, t ∈ [0, τ):H(t) = H_r + I0/k + (H(0) - H_r - I0/k) e^{-kt}Substitute H(0):= H_r + I0/k + [H_r + (I0/k)/(1 + e^{kτ}) - H_r - I0/k] e^{-kt}Simplify inside the brackets:= [ (I0/k)/(1 + e^{kτ}) - I0/k ] = I0/k [1/(1 + e^{kτ}) - 1] = I0/k [ - e^{kτ}/(1 + e^{kτ}) ]So,H(t) = H_r + I0/k - (I0/k) e^{kτ}/(1 + e^{kτ}) e^{-kt}= H_r + I0/k [1 - e^{kτ - kt}/(1 + e^{kτ}) ]= H_r + I0/k [1 - e^{k(τ - t)}/(1 + e^{kτ}) ]Similarly, during rest, t ∈ [τ, 2τ):H(t) = H_r + C e^{-kt}We had C = H(0) - H_r + I0/k (e^{kτ} - 1)Substitute H(0):= [H_r + (I0/k)/(1 + e^{kτ})] - H_r + I0/k (e^{kτ} - 1)= (I0/k)/(1 + e^{kτ}) + I0/k (e^{kτ} - 1)Factor I0/k:= I0/k [1/(1 + e^{kτ}) + e^{kτ} - 1]Simplify inside:= I0/k [ (1 + (e^{kτ} - 1)(1 + e^{kτ})) / (1 + e^{kτ}) ]Wait, maybe better to combine terms:= I0/k [ (1 + e^{kτ}(1 + e^{kτ}) - (1 + e^{kτ})) / (1 + e^{kτ}) ]Wait, perhaps another approach:Let me compute 1/(1 + e^{kτ}) + e^{kτ} - 1:= [1 + (e^{kτ} - 1)(1 + e^{kτ})] / (1 + e^{kτ})Wait, maybe it's easier to compute numerically:Let me denote A = e^{kτ}Then,1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, let me compute directly:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, actually, let's compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not correct. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not the way. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, actually, let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not the correct way to combine. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not right. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, I'm getting confused. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not correct. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not the way. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, I think I'm overcomplicating. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not correct. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not the way. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, I think I'm stuck here. Maybe compute numerically:Let me compute 1/(1 + A) + A - 1:= 1/(1 + A) + (A - 1)= [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not correct. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not correct. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not the way. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, I think I'm making a mistake here. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, no, that's not correct. Let me compute:1/(1 + A) + A - 1 = [1 + (A - 1)(1 + A)] / (1 + A)Wait, I think I need to stop and compute it step by step.Let me compute 1/(1 + A) + A - 1:First, compute A - 1:= A - 1Then, add 1/(1 + A):= (A - 1) + 1/(1 + A)To combine these, find a common denominator, which is (1 + A):= [ (A - 1)(1 + A) + 1 ] / (1 + A)Expand the numerator:= [A(1 + A) - 1(1 + A) + 1] / (1 + A)= [A + A^2 - 1 - A + 1] / (1 + A)Simplify:A + A^2 - 1 - A + 1 = A^2So,= A^2 / (1 + A)Therefore,1/(1 + A) + A - 1 = A^2 / (1 + A)So, going back,C = I0/k * A^2 / (1 + A)But A = e^{kτ}, so:C = I0/k * e^{2kτ} / (1 + e^{kτ})Therefore, during rest, H(t) = H_r + C e^{-kt} = H_r + (I0/k) e^{2kτ} / (1 + e^{kτ}) e^{-kt}Simplify:= H_r + (I0/k) e^{2kτ - kt} / (1 + e^{kτ})= H_r + (I0/k) e^{k(2τ - t)} / (1 + e^{kτ})Alternatively, since 2τ = T, and τ = T/2, so 2τ - t = T - t.But maybe it's better to keep it as is.So, summarizing:During exercise (t ∈ [0, τ)):H(t) = H_r + (I0/k) [1 - e^{k(τ - t)}/(1 + e^{kτ}) ]During rest (t ∈ [τ, 2τ)):H(t) = H_r + (I0/k) e^{k(2τ - t)} / (1 + e^{kτ})Alternatively, we can write both solutions in terms of t.But maybe it's better to express it in terms of the period T, since τ = T/2.So, let me substitute τ = T/2:During exercise (t ∈ [0, T/2)):H(t) = H_r + (I0/k) [1 - e^{k(T/2 - t)}/(1 + e^{kT/2}) ]During rest (t ∈ [T/2, T)):H(t) = H_r + (I0/k) e^{k(T - t)} / (1 + e^{kT/2})This should be the steady-state periodic solution over one complete cycle.Alternatively, we can write it in terms of τ:H(t) = H_r + (I0/k) [1 - e^{k(τ - t)}/(1 + e^{kτ}) ] for t ∈ [0, τ)H(t) = H_r + (I0/k) e^{k(2τ - t)}/(1 + e^{kτ}) for t ∈ [τ, 2τ)I think this is the solution. Let me check if it makes sense.At t=0, H(0) = H_r + (I0/k) [1 - e^{kτ}/(1 + e^{kτ}) ] = H_r + (I0/k) [ (1 + e^{kτ} - e^{kτ}) / (1 + e^{kτ}) ] = H_r + (I0/k) [1 / (1 + e^{kτ}) ]Which matches our earlier result.At t=τ-, H(t) approaches H_r + (I0/k) [1 - e^{0}/(1 + e^{kτ}) ] = H_r + (I0/k) [1 - 1/(1 + e^{kτ}) ] = H_r + (I0/k) [ e^{kτ}/(1 + e^{kτ}) ]At t=τ+, H(t) = H_r + (I0/k) e^{k(2τ - τ)}/(1 + e^{kτ}) = H_r + (I0/k) e^{kτ}/(1 + e^{kτ})Which matches, so the solution is continuous at t=τ.Good, that seems consistent.So, part 1 is solved. The steady-state solution is piecewise defined over the exercise and rest intervals as above.Now, moving to part 2: The fitness blogger wants the average heart rate over one period T. So, we need to compute the average H(t) over [0, T).The average heart rate is given by:bar{H} = (1/T) ∫₀^T H(t) dtSince H(t) is periodic with period T, we can compute the integral over one period.Given that H(t) is piecewise defined, we can split the integral into two parts: exercise and rest.So,bar{H} = (1/T) [ ∫₀^{T/2} H(t) dt + ∫_{T/2}^T H(t) dt ]We have expressions for H(t) in both intervals.Let me denote τ = T/2 for simplicity.So,bar{H} = (1/(2τ)) [ ∫₀^τ H(t) dt + ∫_τ^{2τ} H(t) dt ]We can compute each integral separately.First, compute ∫₀^τ H(t) dt where H(t) = H_r + (I0/k) [1 - e^{k(τ - t)}/(1 + e^{kτ}) ]So,∫₀^τ H(t) dt = ∫₀^τ [ H_r + (I0/k) (1 - e^{k(τ - t)}/(1 + e^{kτ}) ) ] dt= H_r τ + (I0/k) ∫₀^τ [1 - e^{k(τ - t)}/(1 + e^{kτ}) ] dtCompute the integral:= H_r τ + (I0/k) [ ∫₀^τ 1 dt - ∫₀^τ e^{k(τ - t)}/(1 + e^{kτ}) dt ]= H_r τ + (I0/k) [ τ - (1/(1 + e^{kτ})) ∫₀^τ e^{k(τ - t)} dt ]Let me compute ∫₀^τ e^{k(τ - t)} dt:Let u = τ - t, then du = -dt, when t=0, u=τ; t=τ, u=0.So,∫₀^τ e^{k(τ - t)} dt = ∫_τ^0 e^{ku} (-du) = ∫₀^τ e^{ku} du = (e^{kτ} - 1)/kTherefore,∫₀^τ H(t) dt = H_r τ + (I0/k) [ τ - (1/(1 + e^{kτ})) (e^{kτ} - 1)/k ]Simplify:= H_r τ + (I0/k) τ - (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ})Similarly, compute ∫_τ^{2τ} H(t) dt where H(t) = H_r + (I0/k) e^{k(2τ - t)}/(1 + e^{kτ})So,∫_τ^{2τ} H(t) dt = ∫_τ^{2τ} [ H_r + (I0/k) e^{k(2τ - t)}/(1 + e^{kτ}) ] dt= H_r τ + (I0/k)/(1 + e^{kτ}) ∫_τ^{2τ} e^{k(2τ - t)} dtCompute the integral:Let u = 2τ - t, then du = -dt, when t=τ, u=τ; t=2τ, u=0.So,∫_τ^{2τ} e^{k(2τ - t)} dt = ∫_τ^0 e^{ku} (-du) = ∫₀^τ e^{ku} du = (e^{kτ} - 1)/kTherefore,∫_τ^{2τ} H(t) dt = H_r τ + (I0/k)/(1 + e^{kτ}) * (e^{kτ} - 1)/k= H_r τ + (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ})Now, sum both integrals:∫₀^{2τ} H(t) dt = [ H_r τ + (I0/k) τ - (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ}) ] + [ H_r τ + (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ}) ]Simplify:= 2 H_r τ + (I0/k) τBecause the last terms cancel out:- (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ}) + (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ}) = 0So,∫₀^{2τ} H(t) dt = 2 H_r τ + (I0/k) τBut τ = T/2, so:= 2 H_r (T/2) + (I0/k) (T/2) = H_r T + (I0 T)/(2k)Therefore, the average heart rate is:bar{H} = (1/T) [ H_r T + (I0 T)/(2k) ] = H_r + I0/(2k)Wait, that's interesting. The average heart rate is simply H_r plus half of I0 divided by k.But let me verify this because it seems too simple. Let me check the calculations.We had:∫₀^{2τ} H(t) dt = 2 H_r τ + (I0/k) τWhich is H_r T + (I0 T)/(2k)So, average is H_r + I0/(2k)Yes, that seems correct.So, regardless of the period T and the duty cycle (as long as the input is periodic with equal exercise and rest durations), the average heart rate is H_r + I0/(2k)Wait, but in our case, the duty cycle is 50%, so the average is H_r + I0/(2k). If the duty cycle were different, say, duty cycle D, then the average would be H_r + D I0/k.But in our case, D=0.5, so it's H_r + I0/(2k)So, for part 2, the average heart rate is H_r + I0/(2k)Given the values: H_r = 60 bpm, k = 0.1, I0 = 30 bpm.So,bar{H} = 60 + 30/(2*0.1) = 60 + 30/0.2 = 60 + 150 = 210 bpmWait, that seems very high. Is that correct?Wait, let me check the units. H_r is in bpm, k is per minute, I0 is in bpm.So, I0/(2k) is (bpm)/(1/min) = bpm*min, which doesn't make sense. Wait, no, wait:Wait, no, the units of k are per minute, so 1/min. So, I0/(2k) is (bpm)/(1/min) = bpm*min, which is not correct for heart rate. Wait, that can't be.Wait, maybe I made a mistake in the units. Let me check the differential equation:dH/dt = -k (H - H_r) + I(t)So, units of dH/dt are bpm per minute (since H is in bpm and t is in minutes). So, k must have units of 1/minute, because k*(H - H_r) must have units of bpm per minute.Similarly, I(t) must have units of bpm per minute, because it's added to dH/dt.Wait, but in the problem statement, I(t) is given as an intensity in bpm. Wait, that might be an issue.Wait, let me re-examine the problem statement.The differential equation is dH/dt = -k (H - H_r) + I(t)H is in bpm, t is in minutes.So, dH/dt is in bpm per minute.Therefore, k must have units of 1/minute, because k*(H - H_r) must have units of bpm per minute.Similarly, I(t) must have units of bpm per minute.But in the problem statement, I(t) is given as a function with I0 in bpm. So, that might be a problem. Because if I(t) is in bpm, then adding it to dH/dt (bpm per minute) is not dimensionally consistent.Wait, that suggests that perhaps I(t) should have units of bpm per minute, or that the equation might have a scaling factor.Wait, maybe the equation should be dH/dt = -k (H - H_r) + I(t), where I(t) is in bpm per minute. So, the units would be consistent.But in the problem statement, I(t) is given as a periodic function with I0 in bpm. So, perhaps there's a scaling factor missing.Alternatively, maybe the equation is correct, and I(t) is in bpm per minute. So, I0 is in bpm per minute.But in the problem statement, it says I0 is 30 bpm. So, that would be inconsistent.Wait, perhaps the equation is actually dH/dt = -k (H - H_r) + I(t), where I(t) is in bpm per minute, but in the problem, I(t) is given as a function with I0 in bpm. So, perhaps there's a scaling factor missing.Alternatively, maybe the equation is correct, and I(t) is in bpm, but then the units don't match. So, perhaps the equation should be dH/dt = -k (H - H_r) + c I(t), where c is a constant to make the units consistent.But since the problem didn't mention that, maybe I should proceed assuming that I(t) is in bpm per minute, so I0 is in bpm per minute.But in the problem statement, it says I0 is 30 bpm. So, perhaps the equation is correct, and I(t) is in bpm per minute, but the problem statement mistakenly says I0 is in bpm.Alternatively, maybe the equation is correct, and I(t) is in bpm, but then the units don't match. So, perhaps the equation should be dH/dt = -k (H - H_r) + I(t), where I(t) is in bpm per minute, but the problem statement says I(t) is in bpm.This is a bit confusing. Let me check the average heart rate expression again.We derived that bar{H} = H_r + I0/(2k)Given H_r = 60 bpm, k = 0.1 per minute, I0 = 30 bpm per minute.Wait, if I0 is in bpm per minute, then I0/(2k) would be (bpm per minute)/(1/minute) = bpm.So, that would make sense.But in the problem statement, I0 is given as 30 bpm, not 30 bpm per minute. So, perhaps there's a scaling factor missing.Alternatively, maybe the equation is correct, and I(t) is in bpm, but then the units don't match. So, perhaps the equation should be dH/dt = -k (H - H_r) + I(t), where I(t) is in bpm per minute, but the problem statement says I(t) is in bpm.This is a bit of a problem. Let me assume that I(t) is in bpm per minute, so I0 = 30 bpm per minute.Then, bar{H} = 60 + 30/(2*0.1) = 60 + 150 = 210 bpmBut 210 bpm is extremely high for a heart rate. The maximum heart rate is around 220 for a young person, so 210 is possible during intense exercise, but it's quite high.Alternatively, maybe I made a mistake in the units. Let me check the differential equation again.dH/dt = -k (H - H_r) + I(t)If H is in bpm, t in minutes, then dH/dt is in bpm per minute.So, k must be in 1/minute, as before.I(t) must be in bpm per minute.But in the problem statement, I(t) is given as a function with I0 in bpm. So, perhaps the equation is actually dH/dt = -k (H - H_r) + c I(t), where c is a constant to convert I(t) from bpm to bpm per minute.But since the problem didn't mention that, perhaps we should proceed with the given values, assuming that I(t) is in bpm per minute, even though the problem says I0 is in bpm.Alternatively, maybe the equation is correct, and I(t) is in bpm, but then the units don't match. So, perhaps the equation should be dH/dt = -k (H - H_r) + I(t), where I(t) is in bpm per minute, but the problem statement mistakenly says I0 is in bpm.Given that, perhaps we should proceed with the calculation as is, even if the units seem inconsistent.So, with H_r = 60, k = 0.1, I0 = 30, T = 10 minutes.So,bar{H} = 60 + 30/(2*0.1) = 60 + 150 = 210 bpmBut as I said, that seems very high. Maybe I made a mistake in the derivation.Wait, let me go back to the integral.We had:∫₀^{2τ} H(t) dt = 2 H_r τ + (I0/k) τBut τ = T/2, so:= H_r T + (I0 T)/(2k)Therefore,bar{H} = (1/T)(H_r T + (I0 T)/(2k)) = H_r + I0/(2k)Yes, that's correct.But if I0 is in bpm, and k is in 1/minute, then I0/(2k) is (bpm)/(1/minute) = bpm*minute, which is not a unit of heart rate.Wait, that suggests that the units are inconsistent, which means that perhaps the equation is missing a scaling factor.Alternatively, maybe the equation is correct, and I(t) is in bpm per minute, so I0 is 30 bpm per minute.In that case, bar{H} = 60 + 30/(2*0.1) = 60 + 150 = 210 bpm, which is possible but very high.Alternatively, maybe the equation is correct, and I(t) is in bpm, but then the units don't match, so perhaps the equation should have I(t) in bpm per minute.Given that, perhaps the correct answer is 210 bpm, but I'm concerned about the units.Alternatively, maybe I made a mistake in the integral.Wait, let me recompute the integrals.First, during exercise:H(t) = H_r + (I0/k) [1 - e^{k(τ - t)}/(1 + e^{kτ}) ]So, integrating from 0 to τ:∫₀^τ H(t) dt = ∫₀^τ [ H_r + (I0/k) (1 - e^{k(τ - t)}/(1 + e^{kτ}) ) ] dt= H_r τ + (I0/k) ∫₀^τ [1 - e^{k(τ - t)}/(1 + e^{kτ}) ] dt= H_r τ + (I0/k) [ τ - (1/(1 + e^{kτ})) ∫₀^τ e^{k(τ - t)} dt ]As before, ∫₀^τ e^{k(τ - t)} dt = (e^{kτ} - 1)/kSo,= H_r τ + (I0/k) τ - (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ})Similarly, during rest:∫_τ^{2τ} H(t) dt = H_r τ + (I0/k^2) (e^{kτ} - 1)/(1 + e^{kτ})Adding them:Total integral = 2 H_r τ + (I0/k) τSo, yes, that's correct.Therefore, the average heart rate is H_r + I0/(2k)Given that, and the values provided, even though the units seem off, I think that's the answer.So, plugging in the numbers:H_r = 60, k = 0.1, I0 = 30bar{H} = 60 + 30/(2*0.1) = 60 + 150 = 210 bpmSo, the average heart rate over one period is 210 bpm.But again, that seems extremely high. Maybe the problem intended I0 to be in a different unit or the equation has a different scaling.Alternatively, perhaps the equation is correct, and I0 is in bpm per minute, so 30 bpm per minute would make sense.In that case, 30 bpm per minute over 5 minutes (half of T=10) would contribute 150 bpm, but that still seems high.Wait, no, the average is over the entire period, so it's H_r plus the average contribution from I(t).Given that, and the math checks out, I think the answer is 210 bpm.So, summarizing:1. The steady-state solution is piecewise defined as above.2. The average heart rate is 210 bpm.</think>