total cost of C(X) = 45,000 + 100x+x^3 dollars. total revenue from the sale of x units is R(X) = 4600x dollars, determine level of production x that will maximize the profit. Find maximum profit.

Question

A manufacturer estimates that its product can be produced at a total cost of C(X) = 45,000 + 100x+x^3 dollars. If the manufacturer's total revenue from the sale of x units is R(X) = 4600x dollars, determine the level of production x that will maximize the profit. Find the maximum profit.

Answer 1

Let P(x) represent the profit, then P=R-C,

P(x)=4600x-(45000+100x+x³)=4500x-45000-x³.

dP/dx=0=4500-3x² at maximum profit, x²=1500, x=38.73 approx. Assuming that x is a quantity then x=39 and profit=4500×39-45000-39³=$71181.