Question

A rectangle has an area of 32 square feet and a perimeter of 24 feet. What is the shortest of the side
lengths, in feet, of the rectangle? F. I
G. 2
Н. 3
J.
4
K. 8

Answer 1

Let the length and width of the rectangle be \( l \) and \( w \), respectively. We know from the problem that:

1. The area of the rectangle is given by:
\[
l \times w = 32
\]

2. The perimeter of the rectangle is given by:
\[
2(l + w) = 24
\]
Simplifying this, we find:
\[
l + w = 12
\]

Now we have a system of equations:
1. \( l \times w = 32 \)
2. \( l + w = 12 \)

We can solve for one variable in terms of the other using the second equation. Let's express \( w \) in terms of \( l \):
\[
w = 12 - l
\]

Next, we substitute this expression for \( w \) into the area equation:
\[
l \times (12 - l) = 32
\]
Expanding this gives:
\[
12l - l^2 = 32
\]
Rearranging this into standard form, we have:
\[
l^2 - 12l + 32 = 0
\]

Now we can solve this quadratic equation using the quadratic formula:
\[
l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In this case, \( a = 1 \), \( b = -12 \), and \( c = 32 \). Plugging in these values, we get:
\[
l = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 32}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
(-12)^2 - 4 \cdot 1 \cdot 32 = 144 - 128 = 16
\]
So the square root of the discriminant is:
\[
\sqrt{16} = 4
\]

Now substituting back into the quadratic formula:
\[
l = \frac{12 \pm 4}{2}
\]
This gives us two possible solutions for \( l \):
1. \( l = \frac{16}{2} = 8 \)
2. \( l = \frac{8}{2} = 4 \)

Now we have two possible lengths: \( l = 8 \) or \( l = 4 \).

To find the corresponding widths:
- If \( l = 8 \), then \( w = 12 - 8 = 4 \).
- If \( l = 4 \), then \( w = 12 - 4 = 8 \).

In both cases, the side lengths of the rectangle are 8 feet and 4 feet. Thus, the shortest side length is:
\[
\text{Shortest side length} = 4 \text{ feet}
\]

Therefore, the answer is \( \boxed{4} \).