Question

A rectangular bathroom mirror has an area of 20 square feet and a perimeter of 18 feet. What are the dimensions of the mirror?

feet by
feet

Answer 1

Let the length of the mirror be \( l \) feet and the width be \( w \) feet. We have the following two equations based on the area and perimeter of the rectangle:

1. The area equation:
\[
l \times w = 20
\]

2. The perimeter equation:
\[
2l + 2w = 18 \quad \text{or simplified,} \quad l + w = 9
\]

Now, we can solve these two equations. From the second equation, we can express \( l \) in terms of \( w \):

\[
l = 9 - w
\]

Now, we substitute this expression for \( l \) into the area equation:

\[
(9 - w) \times w = 20
\]

Expanding this gives:

\[
9w - w^2 = 20
\]

Rearranging the equation, we get a quadratic equation:

\[
w^2 - 9w + 20 = 0
\]

To solve this quadratic equation, we can factor it:

\[
(w - 4)(w - 5) = 0
\]

Setting each factor to zero gives us the possible values for \( w \):

\[
w - 4 = 0 \quad \Rightarrow \quad w = 4
\]
\[
w - 5 = 0 \quad \Rightarrow \quad w = 5
\]

Now, substituting these values back to find \( l \):

If \( w = 4 \):
\[
l = 9 - 4 = 5
\]

If \( w = 5 \):
\[
l = 9 - 5 = 4
\]

Thus, the dimensions of the mirror are \( 4 \) feet by \( 5 \) feet.

So the answer is:

**4 feet by 5 feet**.