Question

A manufacture has been selling 1300 television sets a week at $420 each. A market survey indicates that for each $16 rebate offered to a buyer, the number of sets sold will increase by 160 per week.

If the weekly cost function is 91000+140x, what value of the rebate maximizes the profit? dollars

Answer 1

If y is the number of rebates, then the revenue is

r(y) = (420-16y)(1300+160y)
assuming only enough sets are made to sell out, then x = 1300+160y

so, the profit is revenue less cost

p(y) = r(y) - c(1300+160y)
= (420-16y)(1300+160y) - (91000+140(1300+160y))
= -2560y^2 + 24000y + 273000

This has a maximum at y=75/16

so, a rebate of $75 will maximize the profit

as always, check my math and reasoning.