Question 5 options:

Question

Mary repairs microwaves. Her revenue is modeled by the function
𝑅
(
ℎ
)
=
20
+
30
ℎ
for every h hours she spends repairing microwaves. Her overhead cost is modeled by the function
𝐶
(
ℎ
)
=
10
ℎ
2
−
80
.

After how many hours does she break even?

Enter your answer in the box.

Answer 1

To find out how many hours Mary needs to work to break even, we need to set her revenue equal to her costs:

\[ R(h) = C(h) \]

Given the functions:

\[ R(h) = 20 + 30h \] \[ C(h) = 10h^2 - 80 \]

We set them equal to each other:

\[ 20 + 30h = 10h^2 - 80 \]

Now, rearranging the equation to set it to zero:

\[ 10h^2 - 30h - 100 = 0 \]

Dividing the entire equation by 10 gives:

\[ h^2 - 3h - 10 = 0 \]

Now, we can factor the quadratic equation:

\[ (h - 5)(h + 2) = 0 \]

Setting each factor equal to zero gives us the possible solutions for \( h \):

1. \( h - 5 = 0 \implies h = 5 \)
2. \( h + 2 = 0 \implies h = -2 \) (not a valid solution since hours can't be negative)

Therefore, Mary breaks even after:

\[ \boxed{5} \]

hours.