Question

A car wash firm calculates that its dally profit (in dollars) depends on the number n of workers it employs according to the formula P=−200n+25n2−0.005n4 Calculate the maroinal product at an employment level of 50 workers. HINT [See Example 3.] Internase the result: This means that, ot an employment level of 50 workers, the firm's daily profit will decrease at a rate of $ per additional warker it hires.

Answer 1

To calculate the marginal product at an employment level of 50 workers, we need to find the derivative of the profit function P with respect to the number of workers n, and then substitute n = 50 into the derivative.

Given the profit function: P = -200n + 25n^2 - 0.005n^4

Let's find the derivative of P with respect to n:

dP/dn = d/dn (-200n + 25n^2 - 0.005n^4)

= -200 + 50n - 0.02n^3

Now, substitute n = 50 into the derivative:

dP/dn at n = 50 = -200 + 50 * 50 - 0.02 * 50^3

= -200 + 2500 - 500000

= -497700

So, the marginal product at an employment level of 50 workers is -497700. This means that at an employment level of 50 workers, the firm's daily profit will decrease at a rate of $497700 per additional worker it hires.