he monthly profit for a company that makes decorative picture frames depends on the price per frame. The company determines that the profit is
approximated by f(p) = -80p^2 + 2560p - 17,600, where P is the price per frame and f(p) is the monthly profit based on that price.
(a) Find the price that generates the maximum profit.
(b) Find the maximum profit.
(c) Find the price(s) that would enable the company to break even. If there is more than one price, use the "and" button.
(a) To find the price that generates the maximum profit, we need to find the vertex of the parabola represented by the profit function. The vertex occurs at the price value that maximizes the profit, which can be found using the formula p = -b/(2a), where the profit function is in the form f(p) = ap^2 + bp + c.
In this case, the profit function is f(p) = -80p^2 + 2560p - 17,600.
So, a = -80, b = 2560.
Plug the values into the formula:
p = -2560/(2*(-80))
p = -2560/(-160)
p = 16
Therefore, the price that generates the maximum profit is $16 per frame.
(b) To find the maximum profit, substitute p = 16 into the profit function:
f(16) = -80(16)^2 + 2560(16) - 17,600
f(16) = -80(256) + 40,960 - 17,600
f(16) = -20,480 + 40,960 - 17,600
f(16) = $2,880
Therefore, the maximum profit is $2,880.
(c) To find the price(s) that would enable the company to break even, we need to set the profit function equal to zero:
-80p^2 + 2560p - 17,600 = 0
Divide by -80 throughout:
p^2 - 32p + 220 = 0
Using the quadratic formula to solve for p:
p = [32 ± sqrt((-32)^2 - 4(1)(220))]/(2(1))
p = [32 ± sqrt(1024 - 880)]/2
p = [32 ± √144]/2
p = [32 ± 12]/2
p = 22 or 10
Therefore, the price(s) that would enable the company to break even are $22 per frame and $10 per frame.