To solve the problem, consider the steps below:
1. Define the Variables:
- Let [tex]\( w \)[/tex] be the width of the fence.
- Let [tex]\( l \)[/tex] be the length of the fence.
- According to the problem, the length [tex]\( l \)[/tex] is twice the width [tex]\( w \)[/tex]. Hence, we have the relationship: [tex]\( l = 2w \)[/tex].
2. Understand the Perimeter:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula [tex]\( P = 2 \times (\text{length} + \text{width}) \)[/tex].
- Given that the total perimeter of the fence is 42 feet, we can write the equation: [tex]\( 2 \times (l + w) = 42 \)[/tex].
3. Substitute the Relationship into the Perimeter Equation:
- Since [tex]\( l = 2w \)[/tex], substitute [tex]\( 2w \)[/tex] for [tex]\( l \)[/tex] in the perimeter equation:
[tex]\[
2 \times (2w + w) = 42
\][/tex]
4. Simplify and Solve for [tex]\( w \)[/tex]:
- Combine like terms inside the parentheses:
[tex]\[
2 \times (3w) = 42
\][/tex]
- This simplifies to:
[tex]\[
6w = 42
\][/tex]
- Now, solve for [tex]\( w \)[/tex] by dividing both sides by 6:
[tex]\[
w = \frac{42}{6} = 7
\][/tex]
5. Calculate the Length:
- Given [tex]\( l = 2w \)[/tex], substitute [tex]\( w = 7 \)[/tex]:
[tex]\[
l = 2 \times 7 = 14
\][/tex]
The largest possible length for Tim's fence, given the conditions, is [tex]\( 14 \)[/tex] feet. Checking the options given:
- [tex]\( 6w = 42 ; l=14 \)[/tex] is correct because:
- Here, [tex]\( w = 7 \)[/tex] and [tex]\( l = 14 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{6 w=42 ; l=14} \][/tex]
