Demand and Revenue

Question

A manufacture has been selling 1200 television sets a week at $480 each. A market survey indicates that for each $30 rebate offered to a buyer, the number of sets sold will increase by 300 per week.

a) Find the function representing the demand p(x), where x is the number of the television sets sold per week and p(x) is the corresponding price.

p(x)= 

b) How large rebate should the company offer to a buyer, in order to maximize its revenue? 

c) If the weekly cost function is 96000+160x, how should it set the size of the rebate to maximize its profit? 

Answer 1

P(x)= 600- x/10

R(x) = xP(x) = 600x-x^2/10

R'(x) = MR = 600 -x/5 = 0

x = 5(600)= 3,000

P(3000) = 600-3000/10 = $300 = revenue maximizing price

rebate should be 480-300 = $180

C(x) = 96000+180x

C'(x) = MC = 180

Profit'(x) = R'(x)-C'(x)= MR-MC

= Marginal Revenue minus Marginal Cost

= 600 -x/5 - 180 = 420-x/5 = 0

x/5 = 420

x = 2100

P(2100) = 600-2100/10 = 600-210 = $390 = profit maximizing price, which means the rebate should be 480-390 = $90