A store uses the expression negative 2 p plus 50 to model the number of backpacks it sells per day, where the price, p , can be anywhere from 9 dollars to 15 dollars Which price gives the store the maximum amount of revenue, and what is the maximum revenue? (Revenue is equal to price times number of backpacks.)
Answer
A
12 point 5 0 dollars per backpack gives the maximum revenue; the maximum revenue is 312 point 5 0 dollars .
B
12 point 0 0 dollars per backpack gives the maximum revenue; the maximum revenue is 312 point 0 0 dollars .
C
15 point 0 0 dollars per backpack gives the maximum revenue; the maximum revenue is 20 point 0 0 dollars .
D
9 point 0 0 dollars per backpack gives the maximum revenue; the maximum revenue is 32 point 0 0 .
Explanation
Revenue is price times the number sold, so R(p) = p(-2p + 50) = -2p² + 50p. This downward-opening quadratic reaches its maximum at p = -50/(2 × -2) = $12.50, which is within the allowed range. At that price, the store sells -2(12.5) + 50 = 25 backpacks, so the maximum revenue is $12.50 × 25 = $312.50. Choice A gives this price and revenue.
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