college algebra

Question

A manufacture has been selling 1200 television sets a week at $390 each. A market survey indicates that for each $29 rebate offered to a buyer, the number of sets sold will increase by 290 per week.

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

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 78000+130x78000+130�, how should it set the size of the rebate to maximize its profit?

Answer 1

(1200, 390), (1490, 361) after the $29 rebate with sales up 290

plot them and a line through them.

the line has slope = 290/-29 = (y-390)/(x-1200)

290x -290(1200) =-29p +29(390)

29p = 34800+ 8970 -290x

p = 25830/29 -x

p = 890.69 -x

Revenue = R = px = 890.69x -x^2

R' = 890.69 -2x = 0

x = 890.69/2 = 445.34

p = 890.69 -445.34 = $445.34 = TV revenue maximizing price

Profit = P = R-C

maximum P is when R'-C'= 0, when MR=MC, when Marginal Revenue = Marginal Cost