Questions: The 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)=-80 p^2+2720 p-19,200, 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. Part 1 of 3 (a) The price that generates the maximum profit is 17.00. Alternate Answer: 17 Part: 1 / 3 Part 2 of 3 (b) The maximum profit is 4000.

The 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)=-80 p^2+2720 p-19,200, 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.

Part 1 of 3
(a) The price that generates the maximum profit is 17.00.

Alternate Answer:
17

Part: 1 / 3

Part 2 of 3
(b) The maximum profit is 4000.

Transcript text: The 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 $f(p)=-80 p^{2}+2720 p-19,200$, 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. Part 1 of 3 (a) The price that generates the maximum profit is $\$ 17.00$. Alternate Answer: $\$ 17$ Part: $1 / 3$ Part 2 of 3 (b) The maximum profit is $\$ 4000$. $\square$

Solution

Solution Steps

Step 1: Maximum Profit Price

To find the price that generates the maximum profit, we use the vertex formula for a quadratic function $ f(p) = ap^2 + bp + c $. The price $ p $ that maximizes the profit is given by:

\[ p = -\frac{b}{2a} \]

Substituting $ a = -80 $ and $ b = 2720 $:

\[ p = -\frac{2720}{2 \times -80} = 17.0 \]

Step 2: Maximum Profit Calculation

Next, we calculate the maximum profit by substituting $ p = 17.0 $ back into the profit function:

\[ f(17) = -80(17)^2 + 2720(17) - 19200 \]

Calculating this gives:

\[ f(17) = 3920.0 \]

Step 3: Break-Even Prices

To find the break-even prices, we set the profit function equal to zero:

\[ -80p^2 + 2720p - 19200 = 0 \]

Solving this quadratic equation yields the break-even prices: