How should the cars be priced to maximize profits?

Question

An automobile manufacturer finds that 80,000 cars can be sold if each is priced at $12,000. However, the number sold increases by 25 for every $2 decrease in the price.The manufacturer has fixed costs of $45,000,000; in addition, it costs $5,500 to produce each car. How should the cars be priced to maximize profits

 

(if possible please inform us of break even point, if not it will not matter)

Answer 1

The equation relating sale price S to quantity Q has a slope of -2/25 dollars per car. Therefore S=-(2/25)Q+a where a is calculated by plugging in S=12000 and Q=80000, 12000=-(2/25)80000+a, 12000=-6400+a, a=$18400. Hence S=18400-0.08Q. Revenue R=QS and cost C=45000000+5500Q (fixed cost plus variable cost), profit P=R-C=QS-5500Q-45000000.

P=Q(18400-0.08Q)-5500Q-45000000=

12900Q-0.08Q²-45000000. This has a maximum value of $475,031,250 when Q=12900/0.16=80,625 cars. Therefore the selling price is S=18400-0.08×80625=$11,950.

Breakeven occurs when P=0, 0.08Q²-12900Q+45000000=0.

Q²-161250Q+562500000=0, Q=157,683 or 3,567 cars.