A store has been selling 400 Blu-ray disc players a week at $350 each. A market

Question

A store has been selling 400 Blu-ray disc players a week at $350 each. A market survey indicates that for each $10 rebate offered to buyers, the number of units sold will increase by 20 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue? If x is the number of Blu-ray players sold per week, then the weekly increase in sales is For each increase of 20 units sold, the price is decreased by $10. So for each additional unit sold, the decrease in price will be 1/20 times 10 and the demand function is p(x) = 350 ? (x ? 400) = 550 ? The revenue function is R(x) = xp(x) = Since R'(x) = we see that R'(x) = 0 when x = This value of x gives an absolute maximum by the First Derivative Test (or simply by observing that the graph of R is a parabola that opens downward). The corresponding price is p(550) = and the rebate is 350 ? 275 = Therefore, to maximize revenue, the store should offer a rebate of $.

Explanation / Answer

Solution : If x is the number of blue ray players sold per week, then the weekly increase in the sales is (x - 400). For each increase of 20 units sold, the price is decreased by $10. SO for each additional unit sold, the decrease in price would be (10 x 1/20) and the demand function is

p(x) = 350 - 1/2 * (x- 400) = 550 - 0.5x

Revenue function is

R(x) = xp(x) = x * (550 - 0.5x) = 550 x - 0.5x2

since, R'(x) = 550 - x , we see that R'(x) = 0 , when x = 550 units. The value of x given an absolute maximum by the first derivative test.

p(550) = 550 - 0.5x = 550 - 0.5 * 550 = 275 units

and the rebate is 350- 275 = 75 units . Therefore to maximize revenue, the stoe should offer a rebate of $ 75.

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