Revenue Maximize word Problem?????

Question

A manufacturer has been selling 1150 television sets a week at 510 each. A market survey indicates that for each 14 dollar rebate offered to a buyer, the number of sets sold will increase by 140 per week.

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

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

c) If the weekly cost function is 97750 + 170 x, how should it set the size of the rebate to maximize its profit?

Answer 1

a)

(p - 510)/(x - 1150) = - 14/140 = - 1/10

p - 510 = - 0.1(x - 1150)

p - 510 = - 0.1x + 115

p = 625 - 0.1x

b)

R = px

R = 625x - 0.1x^2

R = - 0.1(x^2 - 6250x + (3125)^2 - 9,765,625)

R = - 0.1(x - 3,125)^2 + $976,562.50

p = $625 - $312.5 = $312.50

rebate = $510 - $312.50 = $197.50 for max revenue

c)

P = 625x - 0.1x^2 - 97,750 - 170 x

P = - 0.1(x^2 - 4550x + 977,500)

P = - 0.1(x^2 - 4550x + 2275^2 - 5,175,625 + 977,500)

P = - 0.1(x - 2275)^2 + $419,812.5)

r = $510 - $625 + $0.10(2275)

r = - $115 + $227.50

rebate = $112.50 for max profit