Sure, let's solve this problem step-by-step.
First, we are given that Stanley is building a rectangular fence using 44 feet of fencing. We also know that the width of the fence is two less than half the length. Let's set up our variables:
- Let [tex]\( l \)[/tex] be the length of the fence.
- Let [tex]\( w \)[/tex] be the width of the fence.
According to the problem, the width [tex]\( w \)[/tex] is two less than half the length. This can be written as:
[tex]\[ w = \frac{l}{2} - 2 \][/tex]
Since the perimeter of the fence is 44 feet, and the perimeter of a rectangle is given by [tex]\( 2(l + w) \)[/tex], we can write:
[tex]\[ 2(l + w) = 44 \][/tex]
We can now substitute the value of [tex]\( w \)[/tex] from the first equation into the perimeter equation:
[tex]\[ 2\left(l + \left(\frac{l}{2} - 2\right)\right) = 44 \][/tex]
Simplify inside the parentheses:
[tex]\[ 2\left(\frac{3l}{2} - 2\right) = 44 \][/tex]
Distribute the 2:
[tex]\[ 3l - 4 = 44 \][/tex]
This is one of the equations given in the options. Next, we solve for [tex]\( l \)[/tex]:
[tex]\[ 3l - 4 = 44 \][/tex]
[tex]\[ 3l = 44 + 4 \][/tex]
[tex]\[ 3l = 48 \][/tex]
[tex]\[ l = 16 \][/tex]
Now, we need to find the width [tex]\( w \)[/tex]. Using the equation [tex]\( w = \frac{l}{2} - 2 \)[/tex]:
[tex]\[ w = \frac{16}{2} - 2 \][/tex]
[tex]\[ w = 8 - 2 \][/tex]
[tex]\[ w = 6 \][/tex]
The width of the fence is [tex]\( w = 6 \)[/tex] feet.
Now, verifying against the original problem choices:
- [tex]\( 3l - 4 = 44 \)[/tex] yields [tex]\( l = 16 \)[/tex].
- Substituting [tex]\( l = 16 \)[/tex] into [tex]\( w = \frac{l}{2} - 2 \)[/tex]:
Therefore, the correct answer is:
[tex]\[ w = 6 \][/tex]
So, the width of the fence is 6 feet.
