Question

A manager estimates the daily output of their factory to be: Q=7x³+x²y+y³ units, where x is the number of hours of skilled labor and y is the number of hours of unskilled labor. Currently, the factory is using 50 hours of skilled labor and 30 hours of unskilled labor. In order for the manager to maintain the current level of output, unskilled labor should be by approximately hours to offset a one-hour increase in skilled labor, rounded to the nearest hundredth of an hour.

Answer 1

To offset a one-hour increase in skilled labor, unskilled labor should be increased by approximately 120 hours.

Step-by-step explanation:

To maintain the current level of output, we need to offset a one-hour increase in skilled labor. The daily output equation is Q = 7x³ + x²y + y³, where x is the number of hours of skilled labor and y is the number of hours of unskilled labor.

Currently, the factory is using 50 hours of skilled labor and 30 hours of unskilled labor.

To find the change in unskilled labor, we take the partial derivative of Q with respect to x and divide it by the partial derivative of Q with respect to y, and multiply by the change in skilled labor.

Plugging in the current values, we get:

(∂Q/∂x)/(∂Q/∂y) * Δx = ((21x² + 2xy) / (3x² + 3y²)) * Δx = ((21(50)² + 2(50)(30)) / (3(50)² + 3(30)²)) * Δx = 120 * Δx.

Therefore, to offset a one-hour increase in skilled labor, unskilled labor should be increased by approximately 120 hours.