Suppose that production of a certain agricultural crop is related to the number of hours of labor x, and the number of acres of the crop y, according to Find the rate of change of the number of hours, with respect to the number of acres.

Question

Suppose that production of a certain agricultural crop is related to the number of hours of labor x, and the number of acres of the crop y, according to  Find the rate of change of the number of hours, with respect to the number of acres. ​

Quizplus · Accepted Answer

To find the rate of change of the number of hours with respect to acres, we need to take the partial derivative of the production function with respect to acres:

∂P/∂y = 2x + 3y

This tells us that if we increase the number of acres by a small amount, dy, the production will increase by approximately (2x + 3y)dy.

Therefore, the rate of change of the number of hours with respect to acres is:

dH/dy = (∂P/∂y) / (∂P/∂x) = (2x + 3y) / (x + 2y)

Substituting x = 12 and y = 8, we get:

dH/dy = (2(12) + 3(8)) / (12 + 2(8)) = 1.6

So the answer is B.

Suppose That Production of a Certain Agricultural Crop Is Related