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 38X - X² and profits of the developer are 28Y - 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 developer's profits. A) H₁ = H₂ = 6. B) H₁ = 14 and H₂ = 6. C) H₁ = 6 and H₂ = 14. D) H₁ = 8 and H₂ = 13. E) H₁ = 13 and H₂ = 17.

Question

Quizplus · Accepted Answer

To find the optimal solution, we need to maximize the total profit, which is the sum of profits of the airport and the developer. For the combined company, the profit function is: 38X - X² + 28Y - Y² - XY To maximize this profit, we need to take partial derivatives with respect to X and Y, and set them equal to zero: d/dX(38X - X² - XY) = 38 - ² - Y = 0 d/dY(28Y - Y² - XY) = 28 - ² - X = 0 Solving these equations simultaneously, we get X = 6 and Y = 14. Therefore, the optimal solution for the combined company is H₁ = 6. For the independent companies, the profit function for the airport is: 38X - X² - XY To maximize this profit, we again take partial derivative with respect to X and set it equal to zero: d/dX(38X - X² - XY) = 38 - ² - Y = 0 Solving for X, we get X = 13. The profit function for the developer is: 28Y - Y² - XY To maximize this profit, we again take partial derivative with respect to Y and set it equal to zero: d/dY(28Y - Y² - XY) = 28 - ² - X = 0 Solving for Y, we get Y = 14. However, since the airport has to pay the developer the total "damages" XY done by the planes to the developer's profits, the actual profit function for the airport is: 38X - X² - XY - Y² Substituting X = 13 and Y = 14, we get the profit for the airport as 38(13) - 13² - 182 = 494 - 13². Therefore, the optimal solution for the independent companies is H₂ = 14. Hence, the answer is choice C: H₁ = 6 and H₂ = 14.

An Airport Is Located Next to a Housing Development