Price Function given to find maximum profit

Question

1. It costs the American Automobile Company $12,000 to produce each automobile, and fixed costs (rent and other expenses that do not depend on the amount of production) are $20,000 per week. The company’s price function is 𝑝𝑝(𝑥𝑥) = 24,000 − 80𝑥𝑥, where p is the price at which exactly x cars will be sold. a) How many cars should be produced each week to maximize profit? b) For what price should they be sold? c) What is the company’s maximum profit?

Answer 1

Step 1. Write down the production and selling cost.

Production cost/week

Pd(x)=12000*x+20000; x is the number of cars.

S(x)=Price* number of cars produced.=24000x-80x^2

Profit P(x)=S(x)-Pd(x)=-80x^2+12000x-20000

step 2: determination of maximum profit.

(a) To determine maximum equate dP/dx=0

dP/dx=-160x+12000=0

solve for x x=12000/160=75 cars/week

step 3 :determination of selling price, production cost and profit.

(b)

S(x=75)=24000-80*75= 18000 $/car

(c)

Total income =18000*75=1,350,000 $/week

Production cost =12000*75+20000= 920,000$/week

Profit=1,350,000 $- 920000 $=430,000 $/week