To find the area of Sherlyn's rectangular garden, we need to use the given information:
1. The perimeter of the garden is 24 feet.
2. The width of the garden is exactly [tex]\(\frac{1}{2}\)[/tex] of its length.
Let's denote the length of the garden as [tex]\( L \)[/tex] and the width as [tex]\( W \)[/tex]. According to the problem:
[tex]\[ W = \frac{1}{2}L \][/tex]
The formula for the perimeter of a rectangle is:
[tex]\[ \text{Perimeter} = 2 \cdot (L + W) \][/tex]
Given the perimeter is 24 feet, we have:
[tex]\[ 24 = 2 \cdot (L + W) \][/tex]
Substitute [tex]\( W \)[/tex] with [tex]\(\frac{1}{2}L \)[/tex]:
[tex]\[ 24 = 2 \cdot (L + \frac{1}{2}L) \][/tex]
Combine like terms inside the parentheses:
[tex]\[ 24 = 2 \cdot \frac{3}{2}L \][/tex]
Simplify the expression:
[tex]\[ 24 = 3L \][/tex]
To find the length [tex]\( L \)[/tex], we solve for [tex]\( L \)[/tex]:
[tex]\[ L = \frac{24}{3} \][/tex]
[tex]\[ L = 8 \][/tex]
Now, we calculate the width [tex]\( W \)[/tex] using [tex]\( W = \frac{1}{2}L \)[/tex]:
[tex]\[ W = \frac{1}{2} \cdot 8 \][/tex]
[tex]\[ W = 4 \][/tex]
Next, we find the area of the rectangle using the formula:
[tex]\[ \text{Area} = L \cdot W \][/tex]
[tex]\[ \text{Area} = 8 \cdot 4 \][/tex]
[tex]\[ \text{Area} = 32 \][/tex]
Therefore, the area of Sherlyn's garden is 32 square feet. The correct answer is:
[tex]\[ \boxed{32} \][/tex]
