To determine the area of Sherlyn's garden, we need to follow these steps:
1. Define Variables and Equations:
- Let [tex]\( L \)[/tex] be the length of the garden.
- Let [tex]\( W \)[/tex] be the width of the garden.
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula [tex]\( P = 2L + 2W \)[/tex].
2. Given Information:
- The perimeter of the garden is [tex]\( 24 \)[/tex] feet.
- The width [tex]\( W \)[/tex] is exactly [tex]\(\frac{1}{2}\)[/tex] of the length [tex]\( L \)[/tex].
3. Express Width in Terms of Length:
- Since [tex]\( W = \frac{1}{2}L \)[/tex].
4. Substitute the Width into the Perimeter Equation:
- We know [tex]\( P = 24 \)[/tex].
- Substitute [tex]\( W = \frac{1}{2}L \)[/tex] into [tex]\( P = 2L + 2W \)[/tex]:
[tex]\[
24 = 2L + 2\left(\frac{1}{2}L\right)
\][/tex]
Simplify inside the parentheses:
[tex]\[
24 = 2L + L
\][/tex]
Combine the terms:
[tex]\[
24 = 3L
\][/tex]
5. Solve for Length [tex]\( L \)[/tex]:
- Divide both sides by 3:
[tex]\[
L = \frac{24}{3} = 8
\][/tex]
Thus, the length [tex]\( L \)[/tex] of the garden is [tex]\( 8 \)[/tex] feet.
6. Calculate the Width [tex]\( W \)[/tex]:
- Since [tex]\( W = \frac{1}{2}L \)[/tex]:
[tex]\[
W = \frac{1}{2} \times 8 = 4
\][/tex]
Thus, the width [tex]\( W \)[/tex] of the garden is [tex]\( 4 \)[/tex] feet.
7. Calculate the Area of the Garden:
- The area [tex]\( A \)[/tex] of a rectangle is given by [tex]\( A = L \times W \)[/tex]:
[tex]\[
A = 8 \times 4 = 32
\][/tex]
Hence, the area of Sherlyn's garden is [tex]\( 32 \)[/tex] square feet.
