🧠:<think>Alright, so I have this problem about a community theater director planning a new play. The director wants to maximize the audience size while keeping the production budget low. There are two parts to this problem. Let me tackle them one by one.Starting with the first part: Formulating a revenue function R(p) based on the ticket price p. The theater has a seating capacity of 300, and the base ticket price is 10, which sells out all seats. For every 1 increase in ticket price, ticket sales drop by 5%. I need to find the function R(p) and then determine the ticket price that maximizes revenue, along with the maximum revenue.Okay, so let's break this down. Revenue is generally the product of the price per ticket and the number of tickets sold. Here, the price is p, and the number of tickets sold depends on p. The base price is 10, selling 300 tickets. For each 1 increase, sales drop by 5%. So, if the price increases by 1, the number of tickets sold becomes 300 - 5% of 300, which is 300 - 15 = 285. If it increases by another 1, it drops by another 5%, so 285 - 14.25 = 270.75, but since we can't sell a fraction of a ticket, maybe we need to think in terms of percentages rather than absolute numbers.Wait, actually, the problem says "ticket sales will drop by 5% for every 1 increase in ticket price." So, it's a percentage decrease, not a fixed number. So, if the price is p dollars, the number of tickets sold is 300 multiplied by (1 - 0.05*(p - 10)). Because for each dollar above 10, we lose 5% of the tickets.Let me write that down:Number of tickets sold, N(p) = 300 * (1 - 0.05*(p - 10))Simplify that:N(p) = 300 * (1 - 0.05p + 0.5)N(p) = 300 * (1.5 - 0.05p)Wait, that doesn't seem right. Wait, 1 - 0.05*(p - 10) is 1 - 0.05p + 0.5, which is 1.5 - 0.05p. Hmm, but if p is 10, N(p) should be 300. Plugging p=10 into 1.5 - 0.05*10 = 1.5 - 0.5 = 1.0, so N(p)=300*1.0=300. That works. If p=11, N(p)=300*(1.5 - 0.05*11)=300*(1.5 - 0.55)=300*0.95=285. That also works. So, that seems correct.So, N(p) = 300*(1.5 - 0.05p). Alternatively, we can write it as N(p) = 300*(1 - 0.05(p - 10)).But let me think again. If the price increases by 1, the number of tickets sold decreases by 5%. So, the number of tickets sold is 300*(1 - 0.05)^(p - 10). Wait, that might be another way to model it, but the problem says "drop by 5% for every 1 increase." So, it's a linear decrease, not exponential. So, it's a linear relationship, not exponential decay.Therefore, the number of tickets sold is linear in p. So, my initial approach is correct. So, N(p) = 300*(1 - 0.05*(p - 10)).So, revenue R(p) is p * N(p) = p * 300*(1 - 0.05*(p - 10)).Let me compute that:R(p) = 300p*(1 - 0.05p + 0.5)Wait, no, let me compute it correctly.Wait, 1 - 0.05*(p - 10) = 1 - 0.05p + 0.5 = 1.5 - 0.05p. So, R(p) = 300p*(1.5 - 0.05p).Alternatively, let me compute it step by step.R(p) = p * [300*(1 - 0.05*(p - 10))]= 300p*(1 - 0.05p + 0.5)Wait, that's not correct. Wait, 1 - 0.05*(p - 10) is 1 - 0.05p + 0.5, which is 1.5 - 0.05p. So, R(p) = 300p*(1.5 - 0.05p).Let me compute that:R(p) = 300p*(1.5 - 0.05p)= 300*(1.5p - 0.05p^2)= 450p - 15p^2So, R(p) = -15p^2 + 450p.That's a quadratic function in terms of p, opening downward, so the maximum is at the vertex.The vertex of a quadratic function ax^2 + bx + c is at x = -b/(2a). So, here, a = -15, b = 450.So, p = -450/(2*(-15)) = -450/(-30) = 15.So, the ticket price that maximizes revenue is 15.Then, the maximum revenue is R(15) = -15*(15)^2 + 450*15.Compute that:15^2 = 225-15*225 = -3375450*15 = 6750So, R(15) = -3375 + 6750 = 3375.So, maximum revenue is 3375.Wait, but let me double-check the number of tickets sold at p=15.N(15) = 300*(1.5 - 0.05*15) = 300*(1.5 - 0.75) = 300*0.75 = 225 tickets.So, revenue is 15*225 = 3375. That matches.Okay, so part 1 seems done.Now, part 2: The director wants to explore cost-effective set design options. The cost function is C(x) = 2000 + 150x - 10x^2, where x is the number of innovative set design ideas implemented. We need to find the number of ideas x that minimizes the set design cost and determine the minimum cost.Wait, hold on. The function is C(x) = 2000 + 150x - 10x^2. This is a quadratic function in terms of x. Since the coefficient of x^2 is negative (-10), the parabola opens downward, meaning it has a maximum point, not a minimum. But the problem says to find the number of ideas x that minimizes the set design cost. Hmm, that seems contradictory because a downward opening parabola doesn't have a minimum; it goes to negative infinity as x increases. So, perhaps I'm misunderstanding the function.Wait, let me check the function again: C(x) = 2000 + 150x - 10x^2. So, it's a quadratic with a negative leading coefficient, so it's a concave down function, meaning it has a maximum at its vertex, not a minimum. Therefore, the cost function doesn't have a minimum unless x is restricted to a certain domain. But the problem doesn't specify any constraints on x, like x being non-negative or within a certain range. So, if x can be any real number, the cost function will decrease without bound as x increases beyond the vertex, which doesn't make practical sense.Wait, perhaps I misread the function. Let me check again: C(x) = 2000 + 150x - 10x^2. So, it's indeed a concave down function. Therefore, the minimum cost would be at the boundaries of x, but since x is the number of ideas, it must be a non-negative integer. So, perhaps the minimum cost occurs at x=0, but let's see.Wait, if x=0, C(0) = 2000 + 0 - 0 = 2000.If x=1, C(1) = 2000 + 150 - 10 = 2140.x=2: 2000 + 300 - 40 = 2260.x=3: 2000 + 450 - 90 = 2360.x=4: 2000 + 600 - 160 = 2440.x=5: 2000 + 750 - 250 = 2500.x=6: 2000 + 900 - 360 = 2540.x=7: 2000 + 1050 - 490 = 2560.x=8: 2000 + 1200 - 640 = 2560.Wait, at x=8, it's 2560, same as x=7.x=9: 2000 + 1350 - 810 = 2540.x=10: 2000 + 1500 - 1000 = 2500.x=11: 2000 + 1650 - 1210 = 2440.x=12: 2000 + 1800 - 1440 = 2360.x=13: 2000 + 1950 - 1690 = 2260.x=14: 2000 + 2100 - 1960 = 2140.x=15: 2000 + 2250 - 2250 = 2000.Wait, so at x=15, C(x)=2000, same as x=0.So, the cost function is symmetric around the vertex. The vertex is at x = -b/(2a) for the quadratic ax^2 + bx + c. Here, a = -10, b = 150.So, x = -150/(2*(-10)) = -150/(-20) = 7.5.So, the vertex is at x=7.5, which is the maximum point. So, the cost function increases to x=7.5 and then decreases beyond that.But since x must be an integer, the minimum cost would be at the endpoints, either x=0 or x=15, both giving C(x)=2000.Wait, but that seems odd. The problem says "find the number of ideas x that minimizes the set design cost." If x can be any non-negative integer, then the minimum cost is 2000, achieved at x=0 and x=15. But that might not make sense in context because implementing 15 ideas might not be practical, but mathematically, it's possible.Alternatively, perhaps the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would open upwards, having a minimum. But the problem states C(x) = 2000 + 150x - 10x^2. So, unless there's a typo, we have to work with that.Alternatively, maybe the function is C(x) = 2000 + 150x - 10x^2, and x is constrained to be between 0 and some maximum value. But the problem doesn't specify. So, perhaps the minimum cost is 2000, achieved at x=0 and x=15.But let me think again. If x is the number of ideas, it's unlikely to be 15, as that would be a lot. Maybe the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would make sense for a cost function, increasing with x. But since the problem says minus 10x^2, I have to go with that.Alternatively, perhaps the function is C(x) = 2000 + 150x - 10x^2, and x is a continuous variable, but in reality, x must be an integer. So, the minimum cost is 2000 at x=0 or x=15.But that seems counterintuitive because implementing more ideas would decrease the cost? That doesn't make sense. Usually, more ideas would cost more, not less. So, perhaps the function is incorrectly written, or I misread it.Wait, let me check again: "C(x) = 2000 + 150x - 10x^2". So, it's 2000 plus 150x minus 10x squared. So, as x increases, the cost first increases, reaches a maximum at x=7.5, then decreases. So, the minimum cost is at the endpoints, x=0 or x=15, both giving C(x)=2000.But that would mean that implementing 15 ideas brings the cost back down to the original 2000. That seems odd, but mathematically, it's correct.Alternatively, perhaps the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would have a minimum at x = -b/(2a) = -150/(20) = -7.5, which is not feasible, so the minimum would be at x=0, C(x)=2000.But the problem states it's minus 10x^2, so I have to work with that.Therefore, the minimum cost is 2000, achieved when x=0 or x=15.But that seems strange because implementing 15 ideas brings the cost back to the original. Maybe the function is supposed to have a positive coefficient for x^2, but let's proceed as given.So, the number of ideas x that minimizes the set design cost is either 0 or 15, both giving C(x)=2000.But perhaps the problem expects us to find the vertex, even though it's a maximum, but that would be at x=7.5, which is not an integer. So, the minimum cost is at the endpoints.Alternatively, maybe the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would make more sense, having a minimum at x= -150/(2*10)= -7.5, which is not feasible, so the minimum is at x=0, C(x)=2000.But since the problem says minus 10x^2, I have to go with that.Therefore, the minimum cost is 2000, achieved when x=0 or x=15.But let me think again. If x=0, no ideas are implemented, so the cost is 2000. If x=15, 15 ideas are implemented, and the cost is also 2000. So, the function is symmetric around x=7.5, with the maximum at x=7.5.Therefore, the minimum cost is 2000, achieved at x=0 and x=15.But that seems odd because implementing 15 ideas would bring the cost back to the original. Maybe the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would have a minimum at x=0, but that's just a guess.Alternatively, perhaps the function is C(x) = 2000 + 150x - 10x^2, and x is constrained to be between 0 and 15, so the minimum is at x=0 or x=15.But without constraints, the function can go to negative infinity as x increases beyond 15, which doesn't make sense. So, perhaps the problem assumes x is between 0 and 15, making the minimum at x=0 and x=15.Therefore, the number of ideas x that minimizes the set design cost is either 0 or 15, with the minimum cost being 2000.But that seems counterintuitive because implementing more ideas should increase the cost, not decrease it. So, perhaps there's a mistake in the function. Alternatively, maybe the function is correct, and the minimum is at x=0 or x=15.Alternatively, perhaps the function is C(x) = 2000 + 150x - 10x^2, and x is a continuous variable, so the minimum is at x=0 or x approaching infinity, but that doesn't make sense.Wait, no, as x increases beyond 7.5, the cost decreases, but since x can't be negative, the minimum would be at x=0 or x approaching infinity, but in reality, x can't be infinite, so the minimum is at x=0.But in the function, as x approaches infinity, C(x) approaches negative infinity, which is not practical. So, perhaps the problem expects us to consider x as a non-negative integer, and the minimum cost is at x=0, C(x)=2000.Alternatively, maybe the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would have a minimum at x=0, but that's just a guess.Given the problem as stated, I think the answer is that the minimum cost is 2000, achieved when x=0 or x=15.But let me double-check the function:C(x) = 2000 + 150x - 10x^2.So, for x=0: 2000.x=1: 2000 + 150 -10=2140.x=2:2000+300-40=2260.x=3:2000+450-90=2360.x=4:2000+600-160=2440.x=5:2000+750-250=2500.x=6:2000+900-360=2540.x=7:2000+1050-490=2560.x=8:2000+1200-640=2560.x=9:2000+1350-810=2540.x=10:2000+1500-1000=2500.x=11:2000+1650-1210=2440.x=12:2000+1800-1440=2360.x=13:2000+1950-1690=2260.x=14:2000+2100-1960=2140.x=15:2000+2250-2250=2000.So, yes, the cost function is symmetric around x=7.5, with the minimum at x=0 and x=15, both giving C(x)=2000.Therefore, the number of ideas x that minimizes the set design cost is either 0 or 15, with the minimum cost being 2000.But that seems odd because implementing 15 ideas brings the cost back to the original. Maybe the function is supposed to have a positive coefficient for x^2, but I have to go with what's given.So, summarizing:1. Revenue function R(p) = -15p^2 + 450p. The ticket price that maximizes revenue is 15, with maximum revenue of 3375.2. The set design cost function C(x) = 2000 + 150x - 10x^2. The minimum cost is 2000, achieved when x=0 or x=15.But wait, for part 2, the problem says "find the number of ideas x that minimizes the set design cost." So, if x can be any non-negative integer, the minimum is at x=0 or x=15. But in reality, implementing 15 ideas might not be practical, but mathematically, it's correct.Alternatively, if the function is supposed to have a positive x^2 term, the minimum would be at x=0. But as given, it's negative, so the minimum is at x=0 and x=15.Therefore, the answers are:1. Ticket price 15, revenue 3375.2. x=0 or x=15, minimum cost 2000.But perhaps the problem expects us to find the vertex, even though it's a maximum, but that's not a minimum. So, maybe the function is supposed to be C(x) = 2000 + 150x + 10x^2, which would have a minimum at x= -150/(2*10)= -7.5, which is not feasible, so the minimum is at x=0, C(x)=2000.But since the problem states it's minus 10x^2, I have to go with that.So, final answers:1. p = 15, R(p) = 3375.2. x = 0 or 15, C(x) = 2000.</think>

🧠:<think>Alright, so I'm trying to figure out this problem about Australian wrestlers and their performance stats. I have three wrestlers: A, B, and C. Each has a different win probability, and they each compete in 40 matches a year. The first sub-problem is to calculate the expected number of matches won and the variance for each. The second part is a bit more complex, asking for the probability that all three win at least 30 matches each, using the normal approximation to the binomial distribution.Okay, let's start with Sub-problem 1. I remember that for a binomial distribution, the expected value (mean) is n*p, where n is the number of trials and p is the probability of success. So for each wrestler, the expected number of wins should be 40 multiplied by their respective probabilities.For Wrestler A, that would be 40 * 0.75. Let me calculate that: 40 times 0.75 is 30. So Wrestler A is expected to win 30 matches. Similarly, Wrestler B has a probability of 0.65, so 40 * 0.65. Hmm, 40 times 0.6 is 24, and 40 times 0.05 is 2, so total 26. Wrestler B is expected to win 26 matches. Wrestler C has the highest probability at 0.80, so 40 * 0.8 is 32. So Wrestler C is expected to win 32 matches.Now, for the variance. I recall that the variance of a binomial distribution is n*p*(1-p). So, let's compute that for each wrestler.Starting with Wrestler A: 40 * 0.75 * (1 - 0.75). That's 40 * 0.75 * 0.25. Calculating that: 40 * 0.75 is 30, and 30 * 0.25 is 7.5. So the variance is 7.5 for Wrestler A.Wrestler B: 40 * 0.65 * (1 - 0.65). That's 40 * 0.65 * 0.35. Let me compute 40 * 0.65 first, which is 26, then 26 * 0.35. 26 * 0.3 is 7.8, and 26 * 0.05 is 1.3, so total 9.1. So variance is 9.1 for Wrestler B.Wrestler C: 40 * 0.8 * (1 - 0.8). That's 40 * 0.8 * 0.2. 40 * 0.8 is 32, and 32 * 0.2 is 6.4. So variance is 6.4 for Wrestler C.Alright, so that takes care of Sub-problem 1. Each wrestler's expected wins and variances are calculated.Moving on to Sub-problem 2. We need to find the probability that all three wrestlers win at least 30 matches each. The hint is to use the normal approximation to the binomial distribution. I remember that when n is large, the binomial distribution can be approximated by a normal distribution with mean μ and variance σ².So, for each wrestler, we can model their number of wins as a normal distribution with the mean and variance we just calculated. Then, we can find the probability that each is at least 30, and since the matches are independent, we can multiply these probabilities together to get the joint probability.Let me outline the steps:1. For each wrestler, calculate the z-score for 30 wins.2. Find the probability that a normal variable is greater than or equal to that z-score.3. Multiply the three probabilities together to get the final result.Starting with Wrestler A:Mean (μ_A) = 30, Variance (σ²_A) = 7.5, so standard deviation (σ_A) = sqrt(7.5) ≈ 2.7386.We want P(X_A ≥ 30). Since the mean is exactly 30, the z-score is (30 - 30)/2.7386 = 0. So the probability that a normal variable is ≥ 0 is 0.5. But wait, actually, since we're dealing with a discrete distribution approximated by a continuous one, we should apply a continuity correction. So instead of 30, we should use 29.5 as the cutoff.So z_A = (29.5 - 30)/2.7386 ≈ (-0.5)/2.7386 ≈ -0.1826.Looking up this z-score in the standard normal table, the probability that Z ≤ -0.1826 is approximately 0.4292. Therefore, the probability that Z ≥ -0.1826 is 1 - 0.4292 = 0.5708.Wait, hold on. If we're using continuity correction, we want P(X_A ≥ 30) which is equivalent to P(X_A ≥ 29.5) in the continuous approximation. So actually, it's 1 - P(Z ≤ (29.5 - 30)/σ_A). So that would be 1 - Φ(-0.1826) = Φ(0.1826). Φ(0.18) is about 0.5714, so approximately 0.5714.Similarly, for Wrestler B:Mean (μ_B) = 26, Variance (σ²_B) = 9.1, so σ_B ≈ 3.0166.We want P(X_B ≥ 30). Again, applying continuity correction, it's P(X_B ≥ 29.5). So z_B = (29.5 - 26)/3.0166 ≈ 3.5 / 3.0166 ≈ 1.16.Looking up z = 1.16 in the standard normal table, Φ(1.16) ≈ 0.8770. So P(Z ≥ 1.16) = 1 - 0.8770 = 0.1230.Wait, no. Wait, if we have z = (29.5 - 26)/3.0166 ≈ 1.16, then P(Z ≤ 1.16) ≈ 0.8770, so P(Z ≥ 1.16) is 0.1230. So that's the probability for Wrestler B.For Wrestler C:Mean (μ_C) = 32, Variance (σ²_C) = 6.4, so σ_C ≈ 2.5298.We want P(X_C ≥ 30). Applying continuity correction, it's P(X_C ≥ 29.5). So z_C = (29.5 - 32)/2.5298 ≈ (-2.5)/2.5298 ≈ -0.9877.Looking up z = -0.9877, Φ(-0.9877) ≈ 0.1611. So P(Z ≥ -0.9877) = 1 - 0.1611 = 0.8389.So now, the probabilities for each wrestler winning at least 30 matches are approximately:- Wrestler A: ~0.5714- Wrestler B: ~0.1230- Wrestler C: ~0.8389Since the match outcomes are independent, the joint probability is the product of these three probabilities.Calculating that: 0.5714 * 0.1230 * 0.8389.First, multiply 0.5714 and 0.1230:0.5714 * 0.1230 ≈ 0.0702.Then, multiply that by 0.8389:0.0702 * 0.8389 ≈ 0.0588.So approximately a 5.88% chance that all three wrestlers win at least 30 matches each.Wait, that seems low, especially considering Wrestler C has a high probability. Let me double-check the calculations.Starting with Wrestler A:z = (29.5 - 30)/2.7386 ≈ -0.1826. Φ(-0.1826) ≈ 0.4292, so 1 - 0.4292 = 0.5708. That seems correct.Wrestler B:z = (29.5 - 26)/3.0166 ≈ 1.16. Φ(1.16) is about 0.8770, so 1 - 0.8770 = 0.1230. Correct.Wrestler C:z = (29.5 - 32)/2.5298 ≈ -0.9877. Φ(-0.9877) ≈ 0.1611, so 1 - 0.1611 = 0.8389. Correct.Multiplying them: 0.5708 * 0.1230 = approx 0.0702. Then 0.0702 * 0.8389 ≈ 0.0588, which is about 5.88%.Hmm, that seems low, but considering Wrestler B has a relatively low probability, it might make sense. Wrestler B is the weakest link here, with only about 12.3% chance, so even though A and C have higher chances, the overall probability is dragged down by B.Alternatively, maybe I should have used continuity correction differently? Wait, for P(X ≥ 30), we use P(X ≥ 29.5). But for some distributions, sometimes people use P(X ≥ 30.5) for the upper tail. Wait, no, continuity correction for P(X ≥ k) is P(X ≥ k - 0.5). So for k=30, it's P(X ≥ 29.5). So I think I did that correctly.Alternatively, perhaps I should have used the exact binomial probabilities instead of the normal approximation, but the problem specifies to use the normal approximation, so I think that's acceptable.So, in conclusion, the probability is approximately 5.88%.Now, for the blog post, I need to write an insightful piece based on these results. I should probably highlight the expected wins, the variances, and then discuss the low probability of all three having such successful years. It's a good way to show how statistics can inform us about the likelihood of certain outcomes, even for top performers.I should make sure to explain the concepts clearly, maybe touch on what the variance means in terms of consistency, and how the normal approximation helps in calculating joint probabilities. Also, it's important to note that while Wrestler C has the highest chance, Wrestler B is the limiting factor here, significantly lowering the overall probability.I think that's a good outline. Now, I'll proceed to write the blog post incorporating these calculations and insights.</think>

🧠:<think>Okay, so I have this problem about a front-end web developer working on responsive design for iOS devices. The goal is to maintain the golden ratio aspect of 1.618:1. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1: Define the function ( f(x, y) = ax^2 + bxy + cy^2 ) which represents the curvature of a responsive design element. The developer wants this function to remain constant along the curve defined by the golden aspect ratio. Also, the line ( y = 1.618x ) is tangent to this curve at a specific point ((x_0, y_0)). I need to find the relationship between the constants ( a, b, ) and ( c ).Hmm, okay. So, the function ( f(x, y) ) is supposed to be constant along the curve where ( y = 1.618x ). That means for any point on this line, ( f(x, y) ) should have the same value. Let me denote the golden ratio as ( phi = 1.618 ), so ( y = phi x ).So, substituting ( y = phi x ) into ( f(x, y) ), we get:[ f(x, phi x) = a x^2 + b x (phi x) + c (phi x)^2 ]Simplify that:[ f(x, phi x) = a x^2 + b phi x^2 + c phi^2 x^2 ]Factor out ( x^2 ):[ f(x, phi x) = (a + b phi + c phi^2) x^2 ]Since ( f(x, y) ) is constant along this curve, the expression ( (a + b phi + c phi^2) x^2 ) must be constant for all ( x ). The only way this can happen is if the coefficient of ( x^2 ) is zero, because otherwise, as ( x ) changes, the value of ( f ) would change. So, setting the coefficient equal to zero:[ a + b phi + c phi^2 = 0 ]That's one equation relating ( a, b, ) and ( c ).But wait, the problem also mentions that the line ( y = phi x ) is tangent to the curve defined by ( f(x, y) = k ) (where ( k ) is a constant) at the point ( (x_0, y_0) ). So, tangency implies that at that specific point, the gradient of ( f ) is parallel to the gradient of the line ( y = phi x ).Let me recall that the gradient of ( f ) is given by:[ nabla f = left( frac{partial f}{partial x}, frac{partial f}{partial y} right) = (2a x + b y, b x + 2c y) ]The line ( y = phi x ) can be written as ( y - phi x = 0 ), so its gradient is:[ nabla (y - phi x) = (-phi, 1) ]For the gradients to be parallel, there must exist a scalar ( lambda ) such that:[ nabla f = lambda nabla (y - phi x) ]So, at the point ( (x_0, y_0) ), we have:[ 2a x_0 + b y_0 = -lambda phi ][ b x_0 + 2c y_0 = lambda ]Since ( y_0 = phi x_0 ), we can substitute that into the equations:First equation:[ 2a x_0 + b (phi x_0) = -lambda phi ]Factor out ( x_0 ):[ x_0 (2a + b phi) = -lambda phi ]Second equation:[ b x_0 + 2c (phi x_0) = lambda ]Factor out ( x_0 ):[ x_0 (b + 2c phi) = lambda ]Now, let's denote ( x_0 ) as non-zero (since if ( x_0 = 0 ), the point would be at the origin, which might not be meaningful here). So, we can solve for ( lambda ) from both equations.From the first equation:[ lambda = -frac{x_0 (2a + b phi)}{phi} ]From the second equation:[ lambda = x_0 (b + 2c phi) ]Set them equal:[ -frac{x_0 (2a + b phi)}{phi} = x_0 (b + 2c phi) ]Divide both sides by ( x_0 ) (since ( x_0 neq 0 )):[ -frac{2a + b phi}{phi} = b + 2c phi ]Multiply both sides by ( phi ) to eliminate the denominator:[ -(2a + b phi) = b phi + 2c phi^2 ]Distribute the negative sign:[ -2a - b phi = b phi + 2c phi^2 ]Bring all terms to one side:[ -2a - b phi - b phi - 2c phi^2 = 0 ]Combine like terms:[ -2a - 2b phi - 2c phi^2 = 0 ]Divide both sides by -2:[ a + b phi + c phi^2 = 0 ]Wait a second, that's the same equation I got earlier from the function being constant along the curve. So, this doesn't give me any new information. Hmm.So, from both the function being constant and the tangency condition, I end up with the same equation:[ a + b phi + c phi^2 = 0 ]Therefore, the relationship between ( a, b, ) and ( c ) is:[ a + b phi + c phi^2 = 0 ]Since ( phi = 1.618 ), we can write it as:[ a + 1.618b + (1.618)^2 c = 0 ]Calculating ( (1.618)^2 ) is approximately 2.618, so:[ a + 1.618b + 2.618c = 0 ]But since the problem might expect an exact expression, and knowing that ( phi = frac{1 + sqrt{5}}{2} ), so ( phi^2 = phi + 1 ). Therefore, substituting back:[ a + b phi + c (phi + 1) = 0 ]Which simplifies to:[ a + b phi + c phi + c = 0 ]Factor terms with ( phi ):[ a + c + (b + c) phi = 0 ]Since ( phi ) is irrational, the coefficients of the rational and irrational parts must separately be zero. Therefore:[ a + c = 0 ]and[ b + c = 0 ]So, from ( a + c = 0 ), we get ( a = -c ).From ( b + c = 0 ), we get ( b = -c ).Therefore, ( a = b ). So, all three constants are related as ( a = b = -c ).Wait, let me check that. If ( a + c = 0 ), then ( a = -c ). Similarly, ( b + c = 0 ) implies ( b = -c ). Therefore, ( a = b ). So, the relationship is ( a = b ) and ( c = -a ). So, ( a = b = -c ).Yes, that seems correct. So, the constants must satisfy ( a = b ) and ( c = -a ). Therefore, the relationship is ( a = b ) and ( c = -a ).Moving on to part 2: The developer wants to integrate a dynamic iOS feature that adjusts the layout's complexity based on viewport dimensions. The complexity function is defined as ( g(x, y) = log(x^2 + 2xy + y^2) ). We need to find the critical points and classify their nature when the design maintains the aspect ratio of the golden ratio.First, let's note that maintaining the golden aspect ratio means ( y = phi x ). So, we can substitute ( y = phi x ) into ( g(x, y) ) to express it as a function of a single variable, then find its critical points.But wait, actually, the problem says "when the design maintains the aspect ratio of the golden ratio." So, perhaps we need to consider the function ( g(x, y) ) under the constraint ( y = phi x ). So, we can substitute ( y = phi x ) into ( g ) and then find the critical points with respect to ( x ).Alternatively, maybe we need to find the critical points of ( g(x, y) ) subject to the constraint ( y = phi x ). Hmm, but critical points are where the gradient is zero, so perhaps we need to use Lagrange multipliers or substitute the constraint into ( g ) and then find the derivative.Let me think. Since the aspect ratio is fixed, the function ( g ) is being evaluated along the line ( y = phi x ). So, effectively, we can consider ( g ) as a function of one variable ( x ) by substituting ( y = phi x ).So, let's do that substitution:[ g(x, phi x) = log(x^2 + 2x (phi x) + (phi x)^2) ]Simplify inside the log:[ x^2 + 2phi x^2 + phi^2 x^2 = x^2 (1 + 2phi + phi^2) ]So,[ g(x, phi x) = log(x^2 (1 + 2phi + phi^2)) ]Since ( x^2 ) is positive, we can separate the log:[ g(x, phi x) = log(x^2) + log(1 + 2phi + phi^2) ]Which is:[ 2 log|x| + log(1 + 2phi + phi^2) ]But since ( x ) is a viewport width, it's positive, so we can drop the absolute value:[ 2 log x + log(1 + 2phi + phi^2) ]Now, to find critical points, we take the derivative of ( g ) with respect to ( x ) and set it to zero.Compute the derivative:[ frac{d}{dx} g(x, phi x) = frac{2}{x} + 0 ]Because the second term is a constant with respect to ( x ).Set derivative equal to zero:[ frac{2}{x} = 0 ]But ( frac{2}{x} ) is never zero for any finite ( x ). So, does that mean there are no critical points?Wait, that can't be right. Maybe I made a mistake in substitution or differentiation.Wait, let's go back. The function ( g(x, y) = log(x^2 + 2xy + y^2) ). Let me compute the gradient of ( g ) without substituting the constraint.Compute partial derivatives:[ frac{partial g}{partial x} = frac{2x + 2y}{x^2 + 2xy + y^2} ][ frac{partial g}{partial y} = frac{2x + 2y}{x^2 + 2xy + y^2} ]So, the gradient is:[ nabla g = left( frac{2x + 2y}{x^2 + 2xy + y^2}, frac{2x + 2y}{x^2 + 2xy + y^2} right) ]For critical points, we set both partial derivatives equal to zero. So:[ 2x + 2y = 0 ]Which simplifies to:[ x + y = 0 ]So, ( y = -x )But in our case, the viewport dimensions ( x ) and ( y ) are positive (since they are width and height), so ( y = -x ) would imply negative dimensions, which isn't physically meaningful. Therefore, there are no critical points in the domain of positive ( x ) and ( y ).But wait, the problem says "when the design maintains the aspect ratio of the golden ratio." So, perhaps we need to consider the function ( g(x, y) ) under the constraint ( y = phi x ), and find critical points along that line.In that case, as I did earlier, substituting ( y = phi x ) into ( g ), we get ( g(x, phi x) = 2 log x + log(1 + 2phi + phi^2) ). The derivative with respect to ( x ) is ( 2/x ), which is never zero. So, along the line ( y = phi x ), the function ( g ) doesn't have any critical points because its derivative doesn't equal zero anywhere.Alternatively, maybe the problem is asking for critical points of ( g(x, y) ) without the constraint, but considering the aspect ratio. Hmm, but the problem says "when the design maintains the aspect ratio," so I think it's under that constraint.Alternatively, perhaps I'm supposed to consider the function ( g(x, y) ) and find its critical points in the plane, and then see which of those lie on the line ( y = phi x ). But earlier, we saw that the only critical point would be where ( x + y = 0 ), which is not on the line ( y = phi x ) unless ( x = y = 0 ), which is trivial.So, perhaps the conclusion is that there are no critical points when the design maintains the golden aspect ratio.But let me double-check. The function ( g(x, y) = log(x^2 + 2xy + y^2) ). Let's note that ( x^2 + 2xy + y^2 = (x + y)^2 ). So, ( g(x, y) = log((x + y)^2) = 2 log|x + y| ). Since ( x ) and ( y ) are positive, this simplifies to ( 2 log(x + y) ).So, ( g(x, y) = 2 log(x + y) ). The gradient is:[ nabla g = left( frac{2}{x + y}, frac{2}{x + y} right) ]Setting this equal to zero would require ( frac{2}{x + y} = 0 ), which is impossible since ( x + y > 0 ). Therefore, ( g(x, y) ) has no critical points in the domain ( x > 0, y > 0 ).Therefore, when the design maintains the golden aspect ratio, the complexity function ( g(x, y) ) does not have any critical points because the gradient is never zero in the positive quadrant.So, summarizing part 2: There are no critical points for the complexity function ( g(x, y) ) under the golden aspect ratio constraint because the gradient is always positive and never zero in the domain of positive viewport dimensions.Wait, but the problem says "find the critical points and classify their nature." If there are no critical points, then the answer is that there are none. Alternatively, maybe I misinterpreted the problem.Alternatively, perhaps the problem is asking for critical points of ( g ) without the constraint, but considering the aspect ratio as a condition. But as we saw, the only critical point would be at ( x + y = 0 ), which isn't in the positive quadrant.Alternatively, maybe the problem is asking for critical points along the curve defined by the golden ratio, but as we saw, the derivative along that line is ( 2/x ), which is never zero. So, no critical points.Therefore, the conclusion is that there are no critical points for ( g(x, y) ) when the design maintains the golden aspect ratio.Alternatively, perhaps I need to consider the function ( g(x, y) ) with the constraint ( y = phi x ) and use Lagrange multipliers. Let me try that.Let me set up the Lagrangian:[ mathcal{L}(x, y, lambda) = log(x^2 + 2xy + y^2) - lambda (y - phi x) ]Compute the partial derivatives:1. ( frac{partial mathcal{L}}{partial x} = frac{2x + 2y}{x^2 + 2xy + y^2} + lambda phi = 0 )2. ( frac{partial mathcal{L}}{partial y} = frac{2x + 2y}{x^2 + 2xy + y^2} - lambda = 0 )3. ( frac{partial mathcal{L}}{partial lambda} = -(y - phi x) = 0 )From equation 3, we have ( y = phi x ).From equations 1 and 2, we have:From equation 2:[ frac{2x + 2y}{x^2 + 2xy + y^2} = lambda ]From equation 1:[ frac{2x + 2y}{x^2 + 2xy + y^2} + lambda phi = 0 ]Substitute ( lambda ) from equation 2 into equation 1:[ lambda + lambda phi = 0 ][ lambda (1 + phi) = 0 ]Since ( 1 + phi neq 0 ), we have ( lambda = 0 ).But from equation 2, if ( lambda = 0 ), then:[ frac{2x + 2y}{x^2 + 2xy + y^2} = 0 ]Which implies ( 2x + 2y = 0 ), so ( x + y = 0 ). But again, ( x ) and ( y ) are positive, so this is impossible.Therefore, there are no critical points under the constraint ( y = phi x ).So, in conclusion, for part 2, there are no critical points for the complexity function ( g(x, y) ) when the design maintains the golden aspect ratio.Wait, but the problem says "find the critical points and classify their nature." If there are no critical points, then the answer is that there are none. Alternatively, maybe I'm missing something.Alternatively, perhaps the problem is not about constrained optimization but just finding critical points in general, and then considering the aspect ratio. But as we saw, the only critical point would be at ( x + y = 0 ), which isn't in the positive quadrant.Therefore, the final answer for part 2 is that there are no critical points for ( g(x, y) ) under the golden aspect ratio constraint.So, summarizing both parts:1. The relationship between ( a, b, ) and ( c ) is ( a = b = -c ).2. There are no critical points for the complexity function ( g(x, y) ) when the design maintains the golden aspect ratio.Final Answer1. The relationship is ( boxed{a = b = -c} ).2. There are no critical points.</think>

🧠:<think>Alright, so I have this problem where I need to help a client optimize their portfolio involving the USD/ZAR currency pair and Brent crude oil futures. Let me try to break this down step by step.First, the current exchange rate is 15.5, meaning 1 USD equals 15.5 ZAR. The client has 1,000,000 ZAR to invest. They want to hedge against currency risk by investing in Brent crude oil futures. The futures are currently priced at 75 per barrel, and I project a 5% increase in their price over the next quarter. Additionally, the ZAR is expected to appreciate by 2% against the USD.For the first part, I need to figure out how much the investment will be worth in USD after these changes. Let me outline the steps:1. Convert the 1,000,000 ZAR to USD using the current exchange rate.2. Use this USD amount to buy Brent crude oil futures.3. Calculate the new price of Brent crude oil after a 5% increase.4. Determine the value of the futures investment in USD after the price increase.5. Account for the appreciation of ZAR against USD, which affects the initial conversion.Wait, actually, since the entire 1,000,000 ZAR is converted to USD at the current rate, and then invested in Brent crude oil. After the quarter, both the Brent price increases and the ZAR appreciates. So I need to see how these two factors affect the final USD value.Let me compute each step:1. Convert 1,000,000 ZAR to USD: 1,000,000 / 15.5 = approximately 64,516.13 USD.2. Invest this amount in Brent crude oil futures at 75 per barrel. So the number of barrels bought is 64,516.13 / 75 ≈ 860.215 barrels.3. Brent price increases by 5%, so new price is 75 * 1.05 = 78.75 per barrel.4. The value of the futures investment after the increase is 860.215 * 78.75 ≈ let's calculate that: 860.215 * 78.75. Hmm, 800 * 78.75 is 63,000, and 60.215 * 78.75 ≈ 4,739. So total is approximately 63,000 + 4,739 = 67,739 USD.5. Now, the ZAR appreciates by 2% against USD. The initial exchange rate was 15.5 ZAR/USD. A 2% appreciation means the new rate is 15.5 / 1.02 ≈ 15.1961 ZAR/USD.But wait, how does the appreciation affect the final USD value? Since the investment was already converted to USD and then into Brent futures, the appreciation of ZAR affects the initial conversion. Wait, no, because the initial conversion is done at the start, and the appreciation happens over the quarter. So actually, the USD received from the initial conversion will be worth more in ZAR terms, but since the investment is in USD (Brent futures), the final value is in USD, which is then converted back to ZAR? Or is it kept in USD?Wait, the question says "how much will their investment be worth in USD after the appreciation of ZAR and the increase in the price of Brent crude oil?" So the investment is in Brent futures, which is in USD, and the ZAR appreciation affects the initial USD amount.Wait, perhaps I need to consider that the initial 1,000,000 ZAR is converted to USD at 15.5, giving 64,516.13 USD. Then, the Brent futures increase by 5%, so the investment becomes 64,516.13 * 1.05 ≈ 67,741.94 USD. But also, the ZAR appreciates by 2%, so the USD has depreciated in ZAR terms. However, since the investment is in USD, the final value is still in USD, so the appreciation of ZAR doesn't directly affect the USD value of the investment. Wait, but the question says "after the appreciation of ZAR and the increase in the price of Brent crude oil." So perhaps the USD amount is then converted back to ZAR at the new exchange rate?Wait, no, because the investment is in USD (Brent futures), so the final value is in USD. The appreciation of ZAR affects the initial conversion, but since the investment is in USD, the final value is still in USD. So maybe the appreciation of ZAR doesn't directly impact the USD value of the investment, but it affects the initial amount converted.Wait, let me think again. The client starts with 1,000,000 ZAR. They convert it to USD at 15.5, getting 64,516.13 USD. They invest this in Brent futures, which go up by 5%, so they have 64,516.13 * 1.05 = 67,741.94 USD. Now, the ZAR has appreciated by 2%, so the exchange rate is now 15.5 / 1.02 ≈ 15.1961 ZAR/USD. But the investment is in USD, so the final value is 67,741.94 USD. However, if we want to express this in ZAR terms, it would be 67,741.94 * 15.1961 ≈ let's see, 67,741.94 * 15 ≈ 1,016,129.10, plus 67,741.94 * 0.1961 ≈ 13,267. So total ≈ 1,029,396 ZAR. But the question asks for the value in USD, so it's 67,741.94 USD.Wait, but the question says "after the appreciation of ZAR and the increase in the price of Brent crude oil." So perhaps the USD amount is then converted back to ZAR at the new rate, but the question specifies the value in USD, so maybe it's just the 67,741.94 USD.Alternatively, maybe the appreciation of ZAR affects the USD value indirectly because the initial investment was in ZAR. Wait, no, the initial investment was converted to USD, so the USD amount is fixed in terms of ZAR at the initial conversion. The appreciation of ZAR affects the value of the USD in ZAR terms, but since the investment is in USD, the final value is in USD, which is not directly affected by the ZAR appreciation. So perhaps the answer is just 67,741.94 USD.But let me double-check. The client converts 1,000,000 ZAR to USD at 15.5, gets 64,516.13 USD. Invests in Brent, which goes up by 5%, so 64,516.13 * 1.05 = 67,741.94 USD. The ZAR appreciates by 2%, so the USD has depreciated in ZAR terms, but the investment is in USD, so the final value is still 67,741.94 USD. Therefore, the answer is approximately 67,741.94 USD.Now, moving to the second part. The client wants to diversify by investing 60% in Brent futures and 40% in a combination of USD and ZAR. The actual appreciation of ZAR is 1.5%, and Brent increases by 4%.Assumptions: The entire 1,000,000 ZAR is initially converted to USD for the Brent investment. Wait, no, the portfolio is 60% Brent and 40% USD/ZAR. So the initial 1,000,000 ZAR is split into 60% for Brent and 40% kept in USD/ZAR.Wait, but the initial conversion is the entire 1,000,000 ZAR to USD for Brent. Wait, no, the problem says: "The entire 1,000,000 ZAR is initially converted to USD at the current exchange rate for the investment in Brent crude oil futures." So for the first part, it's all in Brent. For the second part, the client considers a different allocation: 60% in Brent and 40% in USD/ZAR. So the initial 1,000,000 ZAR is converted to USD, then 60% is invested in Brent, and 40% is kept in USD and ZAR. But how is the 40% split between USD and ZAR? The problem says "a combination of USD and ZAR," but doesn't specify the exact split. Wait, perhaps the 40% is kept in USD and ZAR in some proportion, but since the client is hedging, maybe they keep some in ZAR and some in USD. But the problem doesn't specify, so maybe the 40% is kept in USD, and the rest is in Brent. Wait, no, the problem says "a combination of USD and ZAR," so perhaps they keep some in USD and some in ZAR, but the exact allocation isn't specified. Hmm, this is a bit unclear.Wait, let me read the problem again: "the client considers investing 60% of their portfolio in Brent crude oil futures and the remaining 40% in a combination of USD and ZAR." So the 40% is split between USD and ZAR, but the exact split isn't given. However, the initial conversion is the entire 1,000,000 ZAR to USD for Brent. Wait, no, in the second part, the portfolio is 60% Brent and 40% USD/ZAR. So the initial 1,000,000 ZAR is converted to USD, then 60% is invested in Brent, and 40% is kept as a combination of USD and ZAR. But how? Maybe the 40% is kept in USD, and the rest is in Brent. But the problem says "combination," so perhaps they keep some in USD and some in ZAR. But without knowing the exact split, it's hard to calculate. Wait, maybe the 40% is kept in USD, and the rest is in Brent. But the problem says "combination," so perhaps they keep some in USD and some in ZAR, but the exact allocation isn't specified. Hmm, this is confusing.Wait, perhaps the 40% is kept in USD, and the rest is in Brent. So the initial 1,000,000 ZAR is converted to USD at 15.5, giving 64,516.13 USD. Then, 60% of the portfolio is Brent, which is 60% of 64,516.13 = 38,709.68 USD. The remaining 40% is 25,806.45 USD, which is kept in USD and ZAR. But how is this 25,806.45 USD split? The problem says "a combination of USD and ZAR," but doesn't specify the ratio. Maybe they keep half in USD and half in ZAR? Or maybe they keep it all in USD? The problem isn't clear. Alternatively, perhaps the 40% is kept in ZAR, but that doesn't make sense because they converted all to USD initially.Wait, perhaps the 40% is kept in USD, and the rest is in Brent. So the 60% is Brent, 40% is USD. So the initial conversion is 1,000,000 ZAR to USD, which is 64,516.13 USD. Then, 60% of this is invested in Brent, which is 38,709.68 USD, and 40% is kept in USD, which is 25,806.45 USD. Then, after the quarter, Brent increases by 4%, and ZAR appreciates by 1.5%.So the Brent investment: 38,709.68 USD * 1.04 = 40,280.07 USD.The USD portion remains 25,806.45 USD, but the ZAR has appreciated by 1.5%, so the USD has depreciated in ZAR terms. Wait, but the USD portion is kept in USD, so its value in USD remains the same, but if we want to express the entire portfolio in USD, we need to consider the USD portion and the Brent portion.Wait, but the USD portion is already in USD, so it remains 25,806.45 USD. The Brent portion is 40,280.07 USD. So the total portfolio in USD is 25,806.45 + 40,280.07 = 66,086.52 USD.But wait, the problem says "the portfolio is rebalanced at the end of the quarter according to the given percentages." So after the quarter, the portfolio is rebalanced to 60% Brent and 40% USD/ZAR. So we need to calculate the value after the changes and then rebalance.Wait, no, the problem says: "if after the quarter, the actual appreciation of ZAR against USD is 1.5% instead of the projected 2%, and the price of Brent crude oil increases by 4% instead of 5%, calculate the final value of the client's portfolio in USD."So the steps are:1. Convert 1,000,000 ZAR to USD at 15.5: 64,516.13 USD.2. Invest 60% in Brent: 60% of 64,516.13 = 38,709.68 USD. Buy Brent at 75 per barrel: 38,709.68 / 75 ≈ 516.13 barrels.3. The remaining 40% is 25,806.45 USD, which is kept in USD and ZAR. But how? The problem says "a combination of USD and ZAR," but doesn't specify the split. Maybe they keep it all in USD, or split it equally. Since it's unclear, perhaps the 40% is kept in USD, so 25,806.45 USD.After the quarter:- Brent increases by 4%, so new price is 75 * 1.04 = 78 USD per barrel. The value of Brent investment is 516.13 * 78 ≈ 40,280.07 USD.- The USD portion remains 25,806.45 USD, but the ZAR has appreciated by 1.5%, so the USD has depreciated in ZAR terms. However, since the USD portion is kept in USD, its value in USD remains the same. But if the portfolio is rebalanced, we need to consider the total value and then reallocate.Wait, the problem says "the portfolio is rebalanced at the end of the quarter according to the given percentages." So after the quarter, the total value is the sum of Brent and USD/ZAR, then rebalanced to 60% Brent and 40% USD/ZAR.But first, let's calculate the total value before rebalancing.Brent value: 40,280.07 USD.USD/ZAR value: 25,806.45 USD. But wait, the USD/ZAR is a combination, so perhaps the 25,806.45 USD is kept in USD, and the rest is in ZAR. Wait, no, the initial 40% is kept in USD and ZAR, but the problem doesn't specify how. This is unclear. Maybe the 40% is kept in USD, so it remains 25,806.45 USD. Alternatively, perhaps the 40% is split into USD and ZAR, but without knowing the split, it's hard to calculate.Alternatively, perhaps the 40% is kept in ZAR, but that doesn't make sense because they converted all to USD initially. Wait, the initial conversion is the entire 1,000,000 ZAR to USD, so the 40% is in USD, and the rest is in Brent. Therefore, the USD portion is 25,806.45 USD, and the Brent is 38,709.68 USD, which becomes 40,280.07 USD after the increase.So total portfolio value is 40,280.07 + 25,806.45 = 66,086.52 USD.But the problem says the portfolio is rebalanced to 60% Brent and 40% USD/ZAR. So we need to take the total value and reallocate.Total value: 66,086.52 USD.60% of this is 66,086.52 * 0.6 = 39,651.91 USD to be in Brent.40% is 66,086.52 * 0.4 = 26,434.61 USD to be in USD/ZAR.But wait, the Brent value after increase is 40,280.07 USD, which is more than the target 39,651.91 USD. So they need to sell some Brent to reduce it to 39,651.91 USD, and the difference is added to the USD/ZAR portion.So the amount to sell is 40,280.07 - 39,651.91 = 628.16 USD worth of Brent. Since Brent is at 78 USD per barrel, the number of barrels to sell is 628.16 / 78 ≈ 8.05 barrels.After selling, the Brent investment is 39,651.91 USD, and the USD/ZAR portion becomes 25,806.45 + 628.16 = 26,434.61 USD.So the final portfolio is 39,651.91 USD in Brent and 26,434.61 USD in USD/ZAR, totaling 66,086.52 USD.But the problem asks for the final value in USD, so it's 66,086.52 USD.Wait, but the USD/ZAR portion is a combination, so perhaps the 26,434.61 USD is split between USD and ZAR. But again, without knowing the exact split, it's unclear. However, since the problem doesn't specify, perhaps we can assume that the USD/ZAR portion is kept in USD, so the final value is just the total in USD.Alternatively, if the USD/ZAR portion is kept in ZAR, we need to convert it back, but the problem asks for the final value in USD, so perhaps it's just the total USD amount.Therefore, the final value is approximately 66,086.52 USD.But let me double-check the calculations.Initial conversion: 1,000,000 ZAR / 15.5 = 64,516.13 USD.60% in Brent: 64,516.13 * 0.6 = 38,709.68 USD. Number of barrels: 38,709.68 / 75 ≈ 516.13 barrels.Brent increases by 4%: 75 * 1.04 = 78 USD. Value: 516.13 * 78 ≈ 40,280.07 USD.40% in USD: 64,516.13 * 0.4 = 25,806.45 USD.Total before rebalancing: 40,280.07 + 25,806.45 = 66,086.52 USD.Rebalance to 60% Brent: 66,086.52 * 0.6 = 39,651.91 USD.So need to sell 40,280.07 - 39,651.91 = 628.16 USD worth of Brent. At 78 USD/barrel, that's 628.16 / 78 ≈ 8.05 barrels.After selling, Brent is 39,651.91 USD, and USD/ZAR is 25,806.45 + 628.16 = 26,434.61 USD.Total remains 66,086.52 USD.So the final value is approximately 66,086.52 USD.But wait, the ZAR has appreciated by 1.5%, so the USD has depreciated in ZAR terms. However, since the USD/ZAR portion is kept in USD, its value in USD remains the same. If we were to express the entire portfolio in ZAR, we would need to convert the USD portions back at the new exchange rate, but the question asks for USD, so it's just 66,086.52 USD.Therefore, the answers are:1. Approximately 67,741.94 USD.2. Approximately 66,086.52 USD.</think>