An airport is located next to a housing development. Where X is the number of planes that land per day and Y is the number of houses in the housing development, profits of the airport are 28X - X² and profits of the developer are 20Y - Y² - XY. Let H₁ be the number of houses built if a single profit-maximizing company owns the airport and the housing development. Let H₂ be the number of houses built if the airport and the housing development are operated independently and the airport has to pay the developer the total "damages" XY done by the planes to the profits of the developer. A) H₁ = H₂ = 4. B) H₁ = 4 and H₂ = 10. C) H₁ = 10 and H₂ = 4. D) H₁ = 6 and H₂ = 9. E) H₁ = 9 and H₂ = 13.

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Quizplus · Accepted Answer

To maximize profits for a single company owning both the airport and the housing development, we need to find the optimal values of X and Y that maximize the combined profit function: P(X,Y) = 28X - X² + 20Y - Y² - XY To maximize this function, we take partial derivatives with respect to both X and Y, and set them equal to zero: ∂P/∂X = 28 - ² = 0 ∂P/∂Y = 20 - ² - X = 0 Solving these equations simultaneously, we get X = 4 and Y = 10. Therefore, H₁ = H₂ = 4, so option A is incorrect. If the airport and the housing development are operated independently, the airport must pay the developer for the damages done to their profits, which is XY. Therefore, to maximize profits, the airport will choose X to be as high as possible while keeping profits positive after paying the developer. Let D be the amount the airport must pay the developer per day, so the airport's profit function becomes: P(X) = 28X - X² - DXY To find the optimal value of X, we set the derivative of P(X) with respect to X equal to zero: ∂P/∂X = 28 - ² - DY = 0 Solving for X, we get X = (28 - D)/². Since X must be an integer, we can see that the largest value of D that allows X to be greater than zero is D = 18, which corresponds to X = 9. When D = 18, the developer's profit is zero, since they are compensated fully for the damages. Therefore, the number of houses built when the airport and the housing development are operated independently is H₂ = 10 - 9 = 1. Option C is incorrect because it has the opposite values for H₁ and H₂. Option D is incorrect because it has values for H₁ and H₂ that are not the same. Option E is incorrect because H₁ and H₂ differ by more than 1.

An Airport Is Located Next to a Housing Development