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 36X - X² and profits of the developer are 42Y - 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 = 18 and H₂ = 20. B)H₁ = 21 and H₂ = 16. C)H₁ = H₂ = 16. D)H₁ = 16 and H₂ = 21. E)H₁ = 20 and H₂ = 24.

Question

Quizplus · Accepted Answer

To find the optimal number of houses for each scenario, we need to maximize the profit function for each option with respect to the number of houses built (H₁ for option 1 and H₂ for option 2): Option 1: Maximize 36X - 42Y for H₁ - We know that Y = H₁ - To find the optimal X value, we need to set the derivative of the profit function with respect to X to 0: d/dX (36X - 42Y) = 36 = 0 X = 18 Therefore, H₁ = Y = 18. Option 2: Maximize 36X - 42(H₂) - X(H₂) for H₂ - We know that Y = H₂ - To find the optimal X value, we need to set the derivative of the profit function with respect to X to 0: d/dX (36X - 42Y - XY) = 36 - Y = 0 Y = 36 Therefore, H₂ = Y = 21. Comparing the two options, we can see that H₁ = 18 and H₂ = 21, so the answer is D.

An Airport Is Located Next to a Housing Development