A manufacturer has been selling 1000 flat-screen TVs a week at $350 each. A mark

Question

A manufacturer has been selling 1000 flat-screen TVs a week at $350 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) = 73,000 + 110x, how should the manufacturer set the size of the rebate in order to maximize its profit?

Explanation / Answer

A manufacturer has been selling 1000 television sets a week at $400 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of sets sold will increase by 80 per week. Round your answers to the nearest dollar. (a) Find the demand function as a function of the dollar value of the rebate(s). p(x)=? (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If the company experiences a cost of $130 per set, how should the manufacturer set the size of the rebate in order to maximize its profit? slope of demand function = change in sales/change in rebate = 80/10 = 8 When rebate = 0, demand = 1000. ? d(r) = 8r + 1000 ----------------- Revenue = demand * (price - rebate) = (8r + 1000)(400 - r) = (8r + 1000)(-r + 400) = -8r² + 2200r + 400,000 Revenue' = -16r + 2200 16r = -2200 r = 137.5 Revenue is maximized when rebate = $138 ---------------------- profit = demand * (price - cost - rebate) = (8r + 1000)(400-130-r) = (8r + 1000)(-r + 270) = -8r² + 1160r + 270,000 profit' = -16r + 1160 r = 1160/16 = 72.5 Profit is maximized when rebate = $73

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