Calc Help Optimization

Question

A company's marketing department has determined that if their product is sold at the price of p dollars per unit, they can sell q=2400−100p units. Each unit costs 3 dollars to make.  

1. What Price, p, should the company charge to maximize their revenue?

2. What price, p, should the company charge to maximize their profits?

Answer 1

Revenue = R = Price x Quantity sold = pq = 2400p - 100p^2

Take the derivative, set equal to zero and solve for p

R' = 2400 -200p = 0

200p =2400

2p =24

p=$12

to maximize profits, you need Profit = Revenue-Cost = 2400p-100p ^2 - 3(2400-100p)

Cost = 3q = 3(2400-100p) Revenue = 2400p - 100p^2

Profit = 2700p - 100p^2 -7200

Take the derivative, set it equal to zero and solve for p

2700 - 200p = 0

p= 2700/200 = $13.50