A manufacturer has been selling 1000 flat-screen TVs a week at $400 each. A mark

Question

A manufacturer has been selling 1000 flat-screen TVs a week at $400 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of TVs sold will increase by 100 per week.

(a) Find the demand function (price p as a function of units sold x).

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

(c) If its weekly cost function is C(x) = 68,000 + 120x, how should the manufacturer set the size of the rebate in order to maximize its profit?

Explanation / Answer

(a)Let revenue=R,    profit=P,   rebate=r, number of sold=x

R = x * (400 - r)

x = 1000 + (100 * r / 10) = 1000 + 10r

The Demand Function = 1000 + 10r

(b)Revenue=R = (1000 + 10r) * (400 - r)

R = 400000 + 4000r - 1000r - 10r^2=400000 + 3000r - 10r^2

For maximum or minimum revenue dR/dr = 3000 - 20r = 0 to

=> 20r = 3000 => r=150

d^2R/dr^2=-20<0 hence maximum at r=150

For maximum revenue, the rebate should be $150

(c) Given cost function C(x) = 68,000 + 120x,

P = Revenue - Cost = 400000 + 3000r - 10r^2- (68000 + 120x)

= 400000 + 3000r - 10r^2 - 68000 - 120(1000 + 10r)=400000 + 3000r - 10r^2 - 68000-120000-1200r=

= 212000 + 1800r - 10r^2

dP/dr = 1800 - 20r = 0

=> 20r = 1800 =>r=1800/20=90

d^2P/dr^2=-20<= hence maximum at r=90

For maximum profit, the rebate should be $90

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