Certainly! To solve this problem, we need to set up and solve an equation based on the given conditions.
Let's designate the length of the fence as [tex]\( l \)[/tex] and the width as [tex]\( w \)[/tex].
1. According to the problem, the width is two less than half the length. We can write this relationship as:
[tex]\[ w = \frac{l}{2} - 2 \][/tex]
2. The total perimeter of the rectangular fence is 44 feet. The perimeter [tex]\( P \)[/tex] of a rectangle is calculated using the formula:
[tex]\[ P = 2l + 2w \][/tex]
Given [tex]\( P = 44 \)[/tex], we have:
[tex]\[ 2l + 2w = 44 \][/tex]
3. Substitute the expression for [tex]\( w \)[/tex] from the first step into the perimeter equation:
[tex]\[ 2l + 2\left(\frac{l}{2} - 2\right) = 44 \][/tex]
4. Simplify the equation:
[tex]\[ 2l + l - 4 = 44 \][/tex]
[tex]\[ 3l - 4 = 44 \][/tex]
5. Add 4 to both sides to isolate the term involving [tex]\( l \)[/tex]:
[tex]\[ 3l - 4 + 4 = 44 + 4 \][/tex]
[tex]\[ 3l = 48 \][/tex]
6. Divide both sides by 3 to solve for [tex]\( l \)[/tex]:
[tex]\[ l = \frac{48}{3} \][/tex]
[tex]\[ l = 16 \][/tex]
So, the length [tex]\( l \)[/tex] is 16 feet.
7. Using the length, find the width using the relationship [tex]\( w = \frac{l}{2} - 2 \)[/tex]:
[tex]\[ w = \frac{16}{2} - 2 \][/tex]
[tex]\[ w = 8 - 2 \][/tex]
[tex]\[ w = 6 \][/tex]
So, the width [tex]\( w \)[/tex] is 6 feet.
Given the equations and solving process, the correct match is:
[tex]\[ 3l - 4 = 44 ; 16 \][/tex]
Therefore, the width of the fence is 6 feet.
