Question

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)= -80p + 3440p -36,000, where p is the price per frame and f(p) is the monthly profit based on that price.

Requried:
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.

Answer 1

Answer:

a. $21.50

b. $980

c. $25 and $18

Step-by-step explanation:

a. The price that generates the maximum profit is

In this question we use the vertex formula i.e shown below:

$(-\frac{b}{2a}, f(-\frac{b}{2a} ))$

where a = -80

b = 3440

c = 36000

hence,

P-coordinate is

$(-\frac{b}{2a}, (-\frac{3440}{2\times -80} ))$

$= \frac{3440}{160}$

= $21.5

b. Now The maximum profit could be determined by the following equation

$f(p) = 80p^2 + 3440p - 36000\\\\f( 21.5) = -80(21.5)^2 + 3440(21.5) - 36000$

= $980

c. The price that would enable the company to break even that is

f(p) = 0

$f(p) = -80p^2 + 3440p - 36000\\\\-80p^2 + 3440p - 36000 = 0\\\\p^2 -43p + 450 = 0\\\\p^2 - 25p - 18p + 450p = 0\\\\p(p - 25) - 18(p-25) = 0\\\\(p - 25) (p - 18) = 0$

By applying the factoring by -50 and then divided it by -80 and after that we split middle value and at last factors could come

(p - 25) = 0 or (p - 18) = 0

so we can write in this form as well which is

p = 25 or p = 18

Therefore the correct answer is $25 and $18