Video Example) EXAMPLE 6 A store has been selling 200 Blu-ray disc players a wee
ID: 2891063 • Letter: V
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Video Example) EXAMPLE 6 A store has been selling 200 Blu-ray disc players a week at $550 each. A market survey indicates that for each $20 rebate offered to buyers, the number of units sold will increase by 40 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue? SOLUTION If x is the number of Blu-ray players sold per week, then the weekly increase in sales is . For each increase of 40 units sold, the price is decreased by $20. So for each additional unit sold, the decrease in price will be x 20 and the demand function is 40 p(x)-550- (x-200) = 650- The revenue function is R(x)-xp(x)- we se that Rwhe. This value of x gives an Since RTx)= x = , absolute maximum by the First Derivative Test (or simply by observing that the graph of R is a parabola that opens downward). The corresponding price is P(650) and the rebate is 550 325 Therefore, to maximize revenue, the store should offer a rebate of Need HelpRead It Talk to a TutorExplanation / Answer
Weekly Increase in sales = Current Sales - Initial Sales = x - 200
Hence, the first blank answer will be (x-200)
Demand Function = 550 - 20/40 * (x - 200) = 550 - 1/2x + 100 = 650 - 0.5x (second blank answer)
Revenue Function = x * Demand Function = x * (650 - 0.5x) = 650x - 0.5x^2
R'(x) = 650 - x = 0, when x = 650
p(650) = 650 - 0.5(650) = 325$
Rebate = Original - p(650) = 550$ - 325$ = 225$
hence the rebate offered is equal to 225$
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