Question

a store uses the expression \\(-2p + 50\\) to model the number of backpacks it sells per day, where the price, \\(p\\), can be anywhere from \\$9 to \\$15. which price gives the store the maximum amount of revenue, and what is the maximum revenue? (revenue = price \\(\times\\) number of backpacks.)

\\$9.00 per backpack gives the maximum revenue; the maximum revenue is \\$32.00.
\\$12.00 per backpack gives the maximum revenue; the maximum revenue is \\$312.00.
\\$12.50 per backpack gives the maximum revenue; the maximum revenue is \\$312.50.
\\$15.00 per backpack gives the maximum revenue; the maximum revenue is \\$20.00.

Explanation:

Define the revenue function

$$ R(p) = p(-2p + 50) = -2p^2 + 50p $$

Find the price for maximum revenue

$$ p = -\frac{b}{2a} = -\frac{50}{2(-2)} = 12.5 $$

Calculate the maximum revenue

$$ R(12.5) = -2(12.5)^2 + 50(12.5) = -2(156.25) + 625 = 312.5 $$

Answer:

• $9.00 per backpack gives the maximum revenue; the maximum revenue is $32.00.
• $12.00 per backpack gives the maximum revenue; the maximum revenue is $312.00.
• $12.50 per backpack gives the maximum revenue; the maximum revenue is $312.50. (Correct answer)
• $15.00 per backpack gives the maximum revenue; the maximum revenue is $20.00.