🧠:<think>Alright, so I've got these two problems to solve, and I need to figure them out step by step. Let's take them one at a time.Problem 1: Average Daily Protein IntakeOkay, the first problem is about calculating the average daily protein intake over the first 12 weeks of a workout program. The protein intake is given by the function ( P(t) = 100 + 20sinleft(frac{pi}{6}tright) ), where ( t ) is the number of weeks since starting the program.Hmm, so I need to find the average protein intake over 12 weeks. Since this is a function of time, I think I need to use calculus here, specifically the concept of the average value of a function over an interval.I remember that the average value of a function ( f(t) ) over the interval ([a, b]) is given by:[text{Average} = frac{1}{b - a} int_{a}^{b} f(t) , dt]In this case, ( a = 0 ) weeks and ( b = 12 ) weeks. So, plugging into the formula, the average protein intake ( overline{P} ) would be:[overline{P} = frac{1}{12 - 0} int_{0}^{12} left(100 + 20sinleft(frac{pi}{6}tright)right) dt]Alright, so I need to compute this integral. Let's break it down into two parts:1. The integral of the constant function 100.2. The integral of ( 20sinleft(frac{pi}{6}tright) ).Starting with the first part:[int_{0}^{12} 100 , dt = 100t Big|_{0}^{12} = 100(12) - 100(0) = 1200]That was straightforward. Now, the second part:[int_{0}^{12} 20sinleft(frac{pi}{6}tright) dt]I need to integrate ( sin(k t) ), which I remember has a standard integral. The integral of ( sin(k t) ) with respect to ( t ) is ( -frac{1}{k}cos(k t) + C ).So, applying that here, where ( k = frac{pi}{6} ):[int 20sinleft(frac{pi}{6}tright) dt = 20 left( -frac{6}{pi} cosleft(frac{pi}{6}tright) right) + C = -frac{120}{pi} cosleft(frac{pi}{6}tright) + C]Now, evaluating this from 0 to 12:[left[ -frac{120}{pi} cosleft(frac{pi}{6} times 12right) right] - left[ -frac{120}{pi} cosleft(frac{pi}{6} times 0right) right]]Simplify the arguments of the cosine:- ( frac{pi}{6} times 12 = 2pi )- ( frac{pi}{6} times 0 = 0 )So, plugging in:[-frac{120}{pi} cos(2pi) - left( -frac{120}{pi} cos(0) right)]I know that ( cos(2pi) = 1 ) and ( cos(0) = 1 ). So:[-frac{120}{pi} times 1 - left( -frac{120}{pi} times 1 right) = -frac{120}{pi} + frac{120}{pi} = 0]Interesting, the integral of the sine function over this interval is zero. That makes sense because the sine function is periodic, and over an integer multiple of its period, the area above and below the x-axis cancels out.So, putting it all together, the total integral is:[1200 + 0 = 1200]Therefore, the average protein intake is:[overline{P} = frac{1}{12} times 1200 = 100 text{ grams per day}]Wait, that seems straightforward. The sine component averages out to zero over the 12 weeks, so the average protein intake is just the constant term, 100 grams per day. That makes sense because the sine function oscillates symmetrically around zero, so its average over a full period is zero.Problem 2: Maximizing Workout EffectivenessAlright, moving on to the second problem. It's about maximizing the effectiveness of a workout based on the number of compound exercises. The effectiveness ( E(x) ) is given by:[E(x) = 5x - frac{1}{3}x^3]We need to find the number of compound exercises ( x ) that maximizes ( E(x) ), and then calculate the maximum effectiveness.This sounds like a calculus optimization problem. To find the maximum, I should take the derivative of ( E(x) ) with respect to ( x ), set it equal to zero, and solve for ( x ). Then, check if it's a maximum using the second derivative test or some other method.First, let's find the first derivative ( E'(x) ):[E'(x) = frac{d}{dx}left(5x - frac{1}{3}x^3right) = 5 - x^2]Wait, hold on. Let me compute that again:- The derivative of ( 5x ) is 5.- The derivative of ( -frac{1}{3}x^3 ) is ( -frac{1}{3} times 3x^2 = -x^2 ).So, yes, ( E'(x) = 5 - x^2 ).To find critical points, set ( E'(x) = 0 ):[5 - x^2 = 0 implies x^2 = 5 implies x = sqrt{5} text{ or } x = -sqrt{5}]Since the number of exercises can't be negative, we discard ( x = -sqrt{5} ). So, the critical point is at ( x = sqrt{5} ).Now, to determine if this critical point is a maximum, we can use the second derivative test.Compute the second derivative ( E''(x) ):[E''(x) = frac{d}{dx}left(5 - x^2right) = -2x]Evaluate ( E''(x) ) at ( x = sqrt{5} ):[E''(sqrt{5}) = -2 times sqrt{5} approx -4.472]Since ( E''(sqrt{5}) < 0 ), the function is concave down at this point, which means it's a local maximum.Therefore, the number of compound exercises that maximizes effectiveness is ( x = sqrt{5} ). But wait, the number of exercises should be an integer, right? Because you can't do a fraction of an exercise. Hmm, the problem doesn't specify whether ( x ) has to be an integer or if it can be a real number. Let me check the problem statement again.It says, "the number of compound exercises ( x )". It doesn't specify, but in reality, you can't do a fraction of an exercise. So, perhaps we need to consider integer values around ( sqrt{5} ) to find which one gives the maximum effectiveness.But wait, maybe the problem allows ( x ) to be any real number, treating it as a continuous variable for the sake of optimization. Since it's a mathematical model, sometimes they allow continuous variables even if in reality they are discrete.But just to be thorough, let's consider both possibilities.First, if ( x ) can be any real number, then the maximum is at ( x = sqrt{5} approx 2.236 ). But since you can't do a fraction of an exercise, we might need to check ( x = 2 ) and ( x = 3 ) to see which gives a higher effectiveness.Let's compute ( E(2) ) and ( E(3) ):- ( E(2) = 5(2) - frac{1}{3}(2)^3 = 10 - frac{8}{3} = 10 - 2.666... = 7.333... )- ( E(3) = 5(3) - frac{1}{3}(3)^3 = 15 - frac{27}{3} = 15 - 9 = 6 )So, ( E(2) approx 7.333 ) and ( E(3) = 6 ). Therefore, ( x = 2 ) gives a higher effectiveness than ( x = 3 ). So, if we have to choose an integer number of exercises, 2 is better than 3.But wait, let's also check ( x = 1 ) and ( x = 4 ) just to be safe:- ( E(1) = 5(1) - frac{1}{3}(1)^3 = 5 - frac{1}{3} approx 4.666 )- ( E(4) = 5(4) - frac{1}{3}(4)^3 = 20 - frac{64}{3} approx 20 - 21.333 = -1.333 )So, ( E(1) ) is lower than ( E(2) ), and ( E(4) ) is negative, which doesn't make sense in this context because effectiveness can't be negative. So, the maximum effectiveness occurs at ( x = 2 ) if we're restricted to integer values.However, if the problem allows ( x ) to be a real number, then the maximum is at ( x = sqrt{5} approx 2.236 ). But since the problem is about the number of exercises, which must be an integer, I think the answer should be ( x = 2 ).But let me double-check the problem statement:"the number of compound exercises ( x )". It doesn't specify whether ( x ) is an integer or not. Hmm. In mathematical terms, unless specified, variables can be real numbers. So, perhaps the answer is ( x = sqrt{5} ), but in practical terms, it's 2.Wait, but the problem says "the number of compound exercises that maximize the effectiveness". So, if we're talking about the mathematical maximum, it's at ( x = sqrt{5} ), but if we're talking about practical application, it's 2.But since this is a math problem, I think they expect the exact value, which is ( sqrt{5} ), and then compute the maximum effectiveness at that point.So, let's proceed with ( x = sqrt{5} ).Compute ( E(sqrt{5}) ):[E(sqrt{5}) = 5sqrt{5} - frac{1}{3}(sqrt{5})^3]Simplify:First, ( (sqrt{5})^3 = (sqrt{5})^2 times sqrt{5} = 5 times sqrt{5} = 5sqrt{5} )So,[E(sqrt{5}) = 5sqrt{5} - frac{1}{3}(5sqrt{5}) = 5sqrt{5} - frac{5sqrt{5}}{3}]Combine like terms:[5sqrt{5} = frac{15sqrt{5}}{3}]So,[frac{15sqrt{5}}{3} - frac{5sqrt{5}}{3} = frac{10sqrt{5}}{3}]Therefore, the maximum effectiveness is ( frac{10sqrt{5}}{3} ).But let me compute this numerically to get a sense:( sqrt{5} approx 2.236 )So,( frac{10 times 2.236}{3} approx frac{22.36}{3} approx 7.453 )Which is slightly higher than ( E(2) approx 7.333 ), which makes sense because ( sqrt{5} approx 2.236 ) is between 2 and 3, and the function is continuous.So, in conclusion, the number of compound exercises that maximize effectiveness is ( x = sqrt{5} ), and the maximum effectiveness is ( frac{10sqrt{5}}{3} ).But just to make sure, let's recap:1. Took the derivative of ( E(x) ) and found critical point at ( x = sqrt{5} ).2. Second derivative test confirmed it's a maximum.3. Evaluated ( E(x) ) at ( x = sqrt{5} ) to find maximum effectiveness.Yes, that seems solid.Final Answer1. The average daily protein intake over the first 12 weeks is boxed{100} grams.2. The number of compound exercises that maximize effectiveness is boxed{sqrt{5}} and the maximum effectiveness is boxed{dfrac{10sqrt{5}}{3}}.</think>

🧠:<think>Okay, so I have this problem about an obstetrician working in a region with limited medical facilities. She's trying to figure out the probability of pregnant women developing a certain complication. The probability is modeled by the function ( P(C) = 1 - e^{-(aN + bS)} ), where N is the lack of nutrition score, S is the environmental stress score, and a and b are constants.Alright, let's break down the first part of the problem. She has found that the average N is 4 and the average S is 3. The constants a and b are given as 0.2 and 0.3 respectively. I need to calculate P(C).So, plugging the values into the formula, I should compute ( P(C) = 1 - e^{-(0.2*4 + 0.3*3)} ).First, let me compute the exponent part: 0.2 times 4 is 0.8, and 0.3 times 3 is 0.9. Adding those together gives 0.8 + 0.9 = 1.7. So the exponent is -1.7.Therefore, ( P(C) = 1 - e^{-1.7} ). I need to calculate ( e^{-1.7} ). I remember that e is approximately 2.71828. So, calculating ( e^{-1.7} ) is the same as 1 divided by ( e^{1.7} ).Let me compute ( e^{1.7} ). I can use a calculator for this, but since I might not have one, I can approximate it. I know that ( e^{1} ) is about 2.718, ( e^{0.7} ) is roughly 2.0138. So multiplying these together, 2.718 * 2.0138 ≈ 5.473. So, ( e^{1.7} ≈ 5.473 ), which means ( e^{-1.7} ≈ 1 / 5.473 ≈ 0.1827 ).Therefore, ( P(C) = 1 - 0.1827 ≈ 0.8173 ). So, approximately an 81.73% chance of developing the complication.Wait, that seems quite high. Let me double-check my calculations. Maybe my approximation of ( e^{1.7} ) was off. Alternatively, I can use the Taylor series expansion for e^x to get a better approximation.The Taylor series for e^x is ( 1 + x + x^2/2! + x^3/3! + x^4/4! + ... ). For x = 1.7, let's compute up to the 4th term.First term: 1Second term: 1.7Third term: (1.7)^2 / 2 = 2.89 / 2 = 1.445Fourth term: (1.7)^3 / 6 = 4.913 / 6 ≈ 0.8188Fifth term: (1.7)^4 / 24 = 8.3521 / 24 ≈ 0.348Adding these up: 1 + 1.7 = 2.7; 2.7 + 1.445 = 4.145; 4.145 + 0.8188 ≈ 4.9638; 4.9638 + 0.348 ≈ 5.3118.So, with four terms, we get approximately 5.3118. Let's add another term for better accuracy.Sixth term: (1.7)^5 / 120 = 14.19857 / 120 ≈ 0.1183.Adding that: 5.3118 + 0.1183 ≈ 5.4301.So, e^1.7 is approximately 5.4301. Therefore, e^{-1.7} ≈ 1 / 5.4301 ≈ 0.1842.Thus, P(C) = 1 - 0.1842 ≈ 0.8158, or about 81.58%. So, my initial approximation was pretty close. So, roughly 81.6%.Wait, but 81.6% seems high. Is that correct? Let me think about the formula. It's 1 minus e to the negative of (aN + bS). So, as N and S increase, the exponent becomes more negative, making e^(-exponent) smaller, so 1 minus a smaller number is closer to 1. So, yes, higher N and S lead to higher probability. So, with N=4 and S=3, which are both positive, the probability is high.Alternatively, maybe the scores are such that higher scores mean worse conditions, so higher N and S are worse, leading to higher probability. So, 81.6% seems plausible given the parameters.Okay, so moving on to part 2. The obstetrician proposes a nutritional intervention that reduces the average N by 25%. So, the original N was 4. A 25% reduction would be 4 * 0.25 = 1. So, the new N is 4 - 1 = 3.So, N' = 3, and S remains 3. So, we need to compute the new probability P(C') = 1 - e^{-(a*N' + b*S)}.Plugging in the numbers: a = 0.2, N' = 3, b = 0.3, S = 3.So, compute the exponent: 0.2*3 + 0.3*3 = 0.6 + 0.9 = 1.5.Therefore, P(C') = 1 - e^{-1.5}.Again, let's compute e^{-1.5}. I know that e^{-1} ≈ 0.3679, and e^{-0.5} ≈ 0.6065. So, e^{-1.5} = e^{-1} * e^{-0.5} ≈ 0.3679 * 0.6065 ≈ 0.2231.Alternatively, using the Taylor series for e^{-1.5}:Wait, actually, e^{-1.5} is the same as 1 / e^{1.5}. So, let's compute e^{1.5}.Compute e^{1.5} using Taylor series:x = 1.5First term: 1Second term: 1.5Third term: (1.5)^2 / 2 = 2.25 / 2 = 1.125Fourth term: (1.5)^3 / 6 = 3.375 / 6 = 0.5625Fifth term: (1.5)^4 / 24 = 5.0625 / 24 ≈ 0.2109Sixth term: (1.5)^5 / 120 = 7.59375 / 120 ≈ 0.06328Seventh term: (1.5)^6 / 720 ≈ 11.390625 / 720 ≈ 0.01582Eighth term: (1.5)^7 / 5040 ≈ 17.0859375 / 5040 ≈ 0.00339Adding these up:1 + 1.5 = 2.52.5 + 1.125 = 3.6253.625 + 0.5625 = 4.18754.1875 + 0.2109 ≈ 4.39844.3984 + 0.06328 ≈ 4.46174.4617 + 0.01582 ≈ 4.47754.4775 + 0.00339 ≈ 4.4809So, e^{1.5} ≈ 4.4809, so e^{-1.5} ≈ 1 / 4.4809 ≈ 0.2232.Therefore, P(C') = 1 - 0.2232 ≈ 0.7768, or about 77.68%.So, after the intervention, the probability decreases from approximately 81.6% to 77.7%. That's a reduction of about 3.9 percentage points.Wait, is that a significant reduction? It depends on the context, but in terms of public health, even a small reduction can have a big impact when scaled up.Alternatively, maybe I should compute the exact value using a calculator for more precision, but since I don't have one, my approximations should suffice.So, summarizing:1. Original probability: approximately 81.6%2. After intervention: approximately 77.7%Therefore, the intervention reduces the probability by about 3.9%.Wait, let me compute the exact exponent again for part 1:aN + bS = 0.2*4 + 0.3*3 = 0.8 + 0.9 = 1.7, correct.And for part 2: aN' + bS = 0.2*3 + 0.3*3 = 0.6 + 0.9 = 1.5, correct.So, the exponents are correct.Alternatively, maybe I can use logarithm tables or more precise approximations, but I think my current approximations are sufficient for this problem.So, final answers:1. P(C) ≈ 0.816 or 81.6%2. P(C') ≈ 0.777 or 77.7%I think that's it.Final Answer1. The probability of developing the complication is boxed{0.816}.2. The new probability after the intervention is boxed{0.777}.</think>

🧠:<think>Okay, so I have this problem about analyzing a network of financial transactions using graph theory and linear algebra. It's divided into two parts. Let me try to tackle each part step by step.Starting with part 1: I need to derive an expression for the total amount of money transferred in the network using matrix operations. The network is represented by a directed graph G = (V, E), where V is the set of officials and E is the set of transactions. Each edge (u, v) has a weight w(u, v) representing the money transferred from u to v. The adjacency matrix A has entries A_ij equal to the weight of the transaction from official i to official j.Hmm, so the adjacency matrix is a square matrix where each entry A_ij tells me how much money official i sent to official j. To find the total amount of money transferred in the entire network, I think I need to sum up all these individual transactions.In matrix terms, the total money transferred would be the sum of all the entries in the adjacency matrix A. But how do I express that using matrix operations?I remember that the trace of a matrix is the sum of its diagonal elements, but that's not exactly what I need here. The trace would only sum A_ii, which are the transactions from an official to themselves, which might not even exist in this context. So, trace isn't the right approach.Instead, I think I need to sum all the elements of the matrix. How can that be done with matrix operations? I recall that if I multiply a matrix by a vector of ones, it sums the rows or columns. Specifically, if I have a vector 1 with all entries equal to 1, then A multiplied by 1 would give me a vector where each entry is the sum of the corresponding row in A. Then, if I take the transpose of 1 and multiply it by that result, I would get the sum of all the entries in A.Let me write that down:Total money transferred = 1^T * A * 1Where 1 is a column vector of ones with the same number of rows as A.Alternatively, since matrix multiplication is associative, I can also write it as (1^T * A) * 1, which is the same thing.So, in matrix notation, the total amount of money transferred is the product of the ones vector transposed, multiplied by the adjacency matrix, multiplied by the ones vector.Let me double-check that. If A is an n x n matrix, then 1 is an n x 1 vector. So, A * 1 is an n x 1 vector where each entry is the sum of the corresponding row in A. Then, 1^T * (A * 1) is a 1 x 1 matrix, which is the sum of all the entries in A. That makes sense because each entry in A is a transaction, so summing all of them gives the total money transferred.Alternatively, I could think about the Frobenius norm, which is the square root of the sum of the squares of the entries, but that's not what we need here. We just need the sum, not the norm.Another way is to use the element-wise sum, but in terms of matrix operations, the expression 1^T * A * 1 is the standard way to compute the total sum.So, I think that's the expression. Let me write it formally:Total money transferred = mathbf{1}^T A mathbf{1}Where mathbf{1} is a column vector of ones.Moving on to part 2: The head of the committee wants to use the eigenvalues of the adjacency matrix A to identify potential corruption. The eigenvalues are given as λ₁, λ₂, ..., λₙ. I need to prove that the sum of the eigenvalues is equal to the trace of the matrix A. Then, based on this, explain how the eigenvalues can be used to detect unusual patterns.First, the proof that the sum of eigenvalues equals the trace of A.I remember that one of the properties of eigenvalues is that the sum of the eigenvalues of a matrix is equal to the trace of that matrix, which is the sum of the diagonal elements. Let me recall why that is.The trace of a matrix is the sum of its diagonal elements. For eigenvalues, if we consider the characteristic equation det(A - λI) = 0, the sum of the eigenvalues is equal to the coefficient of λ^(n-1) term in the characteristic polynomial, which is equal to the trace of A. Wait, actually, the trace is equal to the sum of the eigenvalues, counting algebraic multiplicities.Yes, that's correct. The trace is the sum of the diagonal elements, and it's also equal to the sum of the eigenvalues. So, for any square matrix A, the sum of its eigenvalues (with multiplicity) is equal to the trace of A.So, mathematically, we can write:sum_{i=1}^{n} lambda_i = text{trace}(A)That's the property we need to use.Now, how can the head of the committee use this to detect unusual patterns?Well, the trace of A is the sum of the diagonal elements, which in the context of the adjacency matrix, would be the sum of transactions where an official sent money to themselves, i.e., self-transactions. However, in many financial networks, self-transactions might be rare or non-existent, depending on the context. So, if the trace is significantly large, it might indicate that some officials are involved in self-dealing or transferring money to themselves, which could be a red flag.Alternatively, the trace could be zero if there are no self-transactions, which is common in many transaction networks. If the trace is non-zero, it might indicate something unusual.But wait, the sum of the eigenvalues is equal to the trace, which is the sum of the diagonal elements. So, if the trace is zero, the sum of eigenvalues is zero. If the trace is non-zero, the sum of eigenvalues is non-zero.However, eigenvalues can be positive or negative, depending on the matrix. In the context of financial transactions, the adjacency matrix A has non-negative weights since money can't be transferred in negative amounts. So, all the entries of A are non-negative.But wait, in a directed graph with transactions, the adjacency matrix can have zero or positive weights. So, A is a non-negative matrix. If it's a non-negative matrix, then according to the Perron-Frobenius theorem, it has a dominant eigenvalue that is real and non-negative, and the corresponding eigenvector has non-negative entries.But in this case, the matrix might not be irreducible or primitive, depending on the network structure. However, the trace being the sum of the eigenvalues, which is equal to the sum of the diagonal elements, which in turn is the sum of self-transactions.So, if the trace is zero, that means there are no self-transactions, which is typical. If the trace is non-zero, that means some officials are transferring money to themselves.But how does that help in detecting corruption? Well, if the trace is unexpectedly large, it could indicate that certain officials are siphoning off funds into their own accounts, which would be a form of corruption.Alternatively, looking at the eigenvalues themselves, the largest eigenvalue (the spectral radius) can indicate the strength of the network. If the spectral radius is unusually large, it might indicate a highly connected component or a hub of transactions that could be suspicious.Moreover, the distribution of eigenvalues can provide information about the structure of the network. For example, if the eigenvalues are spread out in a certain way, it might indicate a regular network, whereas a different spread could indicate a more centralized or irregular network, which might be a sign of corruption.Additionally, if the sum of the eigenvalues (which is the trace) is significantly different from what is expected based on the total transactions, it might indicate some anomalies. For instance, if the total money transferred is known, and the trace is much larger or smaller than expected, it could signal irregularities.Wait, but the trace is specifically the sum of self-transactions, so if the total money transferred is the sum of all entries in A, which is 1^T A 1, as we found earlier, and the trace is just the sum of the diagonal entries, which is a part of that total.So, if the trace is a significant portion of the total, that might be unusual. For example, if most of the transactions are self-transfers, that could be a red flag.Alternatively, if the trace is zero, but the total transactions are high, that's normal. But if the trace is high relative to the total, that could indicate something is wrong.Moreover, eigenvalues can also be used in other ways, such as in detecting communities or clusters in the network. If certain eigenvalues are much larger than others, it might indicate the presence of a few key players or hubs in the network, which could be indicative of corruption if those hubs are centralizing too much money.Another thought: the trace being the sum of eigenvalues, if the trace is much larger than the total transactions divided by the number of officials, it might indicate that some officials are involved in many self-transactions, which could be a sign of corruption.Wait, no, the trace is the sum of self-transactions, not the sum of all transactions. So, if the trace is high, it's specifically about self-transfers, not the overall transactions.So, in summary, the head can use the trace (sum of eigenvalues) to check for the presence of self-transactions. If the trace is unexpectedly high, it might indicate corruption. Additionally, looking at the distribution of eigenvalues can provide insights into the network's structure, such as the presence of dominant nodes or unusual connectivity patterns, which could also be signs of corruption.I think that's a reasonable approach. So, to recap:1. The total money transferred is the sum of all entries in A, which can be expressed as 1^T A 1.2. The sum of eigenvalues equals the trace of A, which is the sum of self-transactions. If this sum is unusually high, it could indicate corruption. Additionally, the eigenvalues' distribution can reveal structural anomalies in the network.I think that covers both parts of the problem.</think>

🧠:<think>Okay, so I have this problem about Alex, who is really into skincare and spends a lot of time at the California beaches. Alex is trying to figure out the best time to stay in the sun without damaging their skin. The problem has two parts, and I need to solve both. Let me start with the first one.Problem 1: Derive the expression for T and calculate the maximum safe exposure time.Alright, the problem says that the maximum safe exposure time, T, is inversely proportional to the UV index, U, and directly proportional to the SPF of the sunscreen. Hmm, okay. So, let me recall what proportional means. If something is directly proportional, it means if one goes up, the other goes up too, right? And inversely proportional is the opposite—if one goes up, the other goes down.So, mathematically, if T is directly proportional to SPF and inversely proportional to U, that should translate to an equation where T equals some constant multiplied by SPF divided by U. Let me write that down:T = k * (SPF) / UWhere k is the constant of proportionality. I think that's the expression they want me to derive. So, that's the first part.Now, they give me specific values: SPF is 30 and UV index is 8. But wait, do I have enough information to find T? It seems like I need the constant k. Hmm, the problem doesn't give me a specific value for k, so maybe I'm supposed to express T in terms of k? Or perhaps there's another way.Wait, maybe I'm overcomplicating. The problem says "derive the expression for T," which I've done as T = k * SPF / U. Then, it says "calculate the maximum safe exposure time T." But without knowing k, I can't calculate a numerical value. Maybe I missed something.Let me reread the problem. It says, "the maximum safe exposure time, T (in minutes), to the sun without causing skin damage is inversely proportional to the UV index, U, and directly proportional to the SPF (Sun Protection Factor) of the sunscreen used." So, the relationship is T = k * SPF / U. But to find k, I might need another piece of information, like a known T for specific SPF and U.Wait, the problem doesn't give me another data point. Maybe I'm supposed to assume a standard value for k? Or perhaps it's a standard formula that I should recall.I remember that the formula for the maximum time you can stay in the sun is often given by T = (SPF * 10) / U, but I'm not sure if that's the case here. Wait, let me think. The standard advice is that without sunscreen, you can stay in the sun for about 10 minutes when the UV index is 1, so SPF 1 would give 10 minutes. Then, SPF 30 would give 300 minutes when UV index is 1. But since UV index is 8, it's inversely proportional, so T = (30 * 10) / 8 = 300 / 8 = 37.5 minutes.Wait, so maybe the constant k is 10? Because T = (SPF * 10) / U. That makes sense because without sunscreen (SPF 1), T would be 10 / U. But when U is 1, T is 10 minutes, which is the standard.So, if I use that, then k is 10. Therefore, the expression is T = (10 * SPF) / U.So, plugging in SPF = 30 and U = 8:T = (10 * 30) / 8 = 300 / 8 = 37.5 minutes.Okay, that seems reasonable. So, the maximum safe exposure time is 37.5 minutes.Wait, but the problem didn't specify that k is 10. Maybe I should have asked for more information. But since it's a common formula, I think it's safe to assume that k is 10. So, I'll go with that.Problem 2: Determine the range of indoor temperatures in both environments that Alex finds comfortable.Alright, so Alex prefers a temperature difference no greater than 15°F between indoor and outdoor environments. The beach temperature is 86°F, and the mountain slopes temperature is 45°F. I need to find the range of indoor temperatures that Alex finds comfortable in both places.So, for the beach, the outdoor temperature is 86°F. Alex wants the indoor temperature to be no more than 15°F different. So, the indoor temperature should be between 86 - 15 and 86 + 15.Similarly, for the mountain slopes, the outdoor temperature is 45°F, so the indoor temperature should be between 45 - 15 and 45 + 15.Let me calculate that.For the beach:Lower bound: 86 - 15 = 71°FUpper bound: 86 + 15 = 101°FSo, the indoor temperature should be between 71°F and 101°F.For the mountain slopes:Lower bound: 45 - 15 = 30°FUpper bound: 45 + 15 = 60°FSo, the indoor temperature should be between 30°F and 60°F.Wait, but 30°F seems quite cold. Is that correct? Let me double-check.Yes, if the outdoor temperature is 45°F, and the difference should be no more than 15°F, then the indoor temperature can be as low as 30°F or as high as 60°F. That seems correct.But wait, 30°F is quite chilly. Maybe Alex would prefer a different range? But the problem states the temperature difference, not the absolute temperature. So, regardless of how cold or hot it is outside, the indoor temperature should be within 15°F of that.So, I think my calculations are correct.Summary of Thoughts:1. For the first part, I derived the formula T = (10 * SPF) / U, assuming k is 10 based on standard skincare advice. Plugging in SPF 30 and UV index 8, I got T = 37.5 minutes.2. For the second part, I calculated the indoor temperature ranges by subtracting and adding 15°F to the outdoor temperatures. For the beach, it's 71°F to 101°F, and for the mountain slopes, it's 30°F to 60°F.I think that's it. I don't see any mistakes in my reasoning, but let me just go through it again quickly.For problem 1, the relationship is T = k * SPF / U. Without another data point, I assumed k is 10, which is standard. So, T = (10 * 30) / 8 = 37.5. That makes sense.For problem 2, the difference is 15°F, so for each outdoor temperature, subtract and add 15 to get the indoor range. 86 ±15 is 71-101, and 45 ±15 is 30-60. Yep, that's correct.I think I'm confident with these answers.Final Answer1. The maximum safe exposure time is boxed{37.5} minutes.2. The comfortable indoor temperature ranges are boxed{71^circ text{F} text{ to } 101^circ text{F}} at the beach and boxed{30^circ text{F} text{ to } 60^circ text{F}} at the mountain slopes.</think>