Question

The annual profit for a firm depends upon the number of units produced. the function which describes the relationship between profit P (stated the number of units produced x is

P-0.01x²+5,000x - 25,000

(a) Determine the number of units x which will result in maximum profit. (b) What is the expected maximum profit?

Answer 1

Given Profit-

$P=-0.01x²+5,000x - 25,000$

Differentiate P w.r.t. x-

$P'=-0.02x+5000$

Putting P'=0, we get-

$-0.02x+5000=0\\\Rightarrow x=\dfrac{5000}{0.02}=250000$

Again Differentiating P' w.r.t. x-

$P''=-0.05<0,$

P is maximum at x=250000

(a) The unit which results in the maximum profit is 250000.

(b) The expected maximum profit is

$P=-0.01(250000)^2+5000(250000)-25000\\=-625000000+1250000000-25000\\=624975000$