Answer to Question #157350 in Calculus for Shahnawaz Rehman

Question

The annual profit for the firm depends upon the number of units produced. Specifically the function which describes the relationship between profit P (stated in dollars) and the number of units produced x is

P=-0.01x^2+5000x-25000

a) Determine the number of units x which will result in maximum profit.

b) What is the expected maximum profit?

Answer 1

Given that

$P = -0.01x^{2} + 5000x - 25000$

Then we have that $\frac{dP}{dx} = -0.02x + 5000$

So when $\frac{dP}{dx} = 0$

We have that $-0.02x + 5000 = 0$

$0.02x = 5000$

$x = \frac{5000}{0.02}$

$x = 250000$

So the answer to question (a) is that 250000 will result in the maximum profit

b) So we substitute the value of $x = 250000$ into $P = -0.01x^{2} + 5000x - 25000$

So we have that $P(250000) =$ $-0.01(250000)^{2} + 5000(250000) - 25000$

$P(250000)= 624975000$

The expected maximum profit is 624975000