Question

Mary repairs microwaves. her revenue is modeled by the function r(h)=20+30h for every h hours she spends repairing microwaves. her overhead cost is modeled by the function c(h)=10h2−80 . after how many hours does she break even?

Answer 1

She breaks even in 5 hours

Answer 2

The correct answer is:

5 hours.

Explanation:

To find the break-even point, we set the two functions equal to each other:

20+30h = 10h²-80

Since this is a quadratic equation, we want it set equal to 0 to solve. First we will subtract 20 from each side:

20+30h-20 = 10h²-80-20

30h = 10h²-100

Subtract 30h from each side:

30h-30h = 10h²-100-30h

0 = 10h²-30h-100

We will use the quadratic formula to solve this:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Instead of x, we have h. Our values for a, b and c are:

a = 10

b = -30

c = -100

This gives us:

$h=\frac{--30\pm \sqrt{(-30)^2-4(10)(-100)}}{2(10)} \\ \\=\frac{30\pm \sqrt{900--4000}}{20} \\ \\=\frac{30\pm \sqrt{900+4000}}{20} \\ \\=\frac{30\pm \sqrt{4900}}{20} \\ \\=\frac{30\pm 70}{20}=\frac{30-70}{20}\text{ or }\frac{30+70}{20} \\ \\=\frac{-40}{20}\text{ or }\frac{100}{20}=-2\text{ or }5$

Since a negative number of time makes no sense, we use h = 5.