Question

The total revenue function for a certain product is given by

R=590x dollars, and the total cost function for this product is

C=15,000 +50x + x squared 2 dollars, where x is the number of units of the product that are produced and sold.

a.	Find the profit function.
b.	Find the number of units that gives maximum profit.
c.	Find the maximum possible profit.

Answer 1

The total revenue is R(x) = 590x

& The total cost is, C(x) = 15000 + 50x + x²

So, the profit function is,

P(x) = R(x) - C(x)

= 590x - (15000 + 50x + x²)

= - x² + 540x - 15000

we need to maximize P(x) with respect to x

So, P'(x) = - 2x + 540

&, P"(x) = - 2 < 0

So, for optimum profit, we must have, P'(x) = 0

So, 2x = 540

So, x = 270

So, (b) 270 units are to be sold for maximum profit.

(c) the maximum profit is,

P(270) = 540x270 - 270² - 15000 = 57900

So, maximum possible profit is 57900 dollars