To solve the problem, let's go through it step-by-step:
1. Understanding the Problem:
- Tim wants to build a rectangular fence.
- He has a total of 42 feet of fencing.
- The length of the rectangle should be twice the width.
2. Formulating the Equation:
- Let’s denote the width of the rectangle by [tex]\( w \)[/tex].
- Since the length is twice the width, the length can be represented as [tex]\( 2w \)[/tex].
- The total perimeter of the rectangle (fencing around the yard) is given by the formula for the perimeter of a rectangle:
[tex]\[
\text{Perimeter} = 2(\text{length} + \text{width})
\][/tex]
- Substituting the length and width, we get:
[tex]\[
2(2w + w) = 42
\][/tex]
3. Simplifying the Equation:
- Inside the parentheses, combine the terms:
[tex]\[
2(3w) = 42
\][/tex]
- Simplify further by multiplying:
[tex]\[
6w = 42
\][/tex]
4. Solving for [tex]\( w \)[/tex]:
- To find the width [tex]\( w \)[/tex], divide both sides of the equation by 6:
[tex]\[
w = \frac{42}{6} = 7
\][/tex]
5. Finding the Length:
- Since the length [tex]\( l \)[/tex] is twice the width:
[tex]\[
l = 2w = 2 \times 7 = 14
\][/tex]
Hence, the largest possible length of the fence is 14 feet.
Thus, the correct answer is represented by the equation:
[tex]\[
6w = 42 \text{ ; } l = 14
\][/tex]
