Question

Suppose that the profit (in hundreds of dollars) of a certain firm is approximated by:

P(x, y)=1500+36 x-1.5 x²+120 y-2 y²
where x is the cost of a unit of labor and y is the cost of a unit of goods. Find values of x and y that maximize profit. Find the maximum profit.

Answer 1

Values that maximize the profit function P(x, y) are x=12 and y=30, resulting in a maximum profit of $35,160. The partial derivatives of the profit function with respect to x and y are set to zero to find these values.

Step-by-step explanation:

To find the values of x and y that maximize the given profit function P(x, y)=1500+36x-1.5x²+120y-2y², we need to calculate the partial derivatives and set them equal to zero, then solve the resulting system of equations. The partial derivative with respect to x is 36 - 3x, and with respect to y is 120 - 4y. Setting these equal to zero and solving for x and y give us the points where the profit is potentially maximized. After solving, we find that x=12 and y=30. Plugging these back into the profit function gives a maximum profit of P(12, 30) = 1500 + 36(12) - 1.5(12²) + 120(30) - 2(30²) = 1500 + 432 - 216 + 3600 - 1800 = 3516 hundred dollars or $35,160.