To solve the problem of finding the largest possible length of the fence, let's follow the structured steps:
1. Define the Variables:
- Let the width of the rectangle be [tex]\( w \)[/tex].
- Since Lance wants the length to be three times the width, the length [tex]\( l \)[/tex] can be defined as [tex]\( l = 3w \)[/tex].
2. Use the Perimeter Constraint:
- The perimeter of a rectangle is calculated using the formula: [tex]\( 2 \times \text{length} + 2 \times \text{width} \)[/tex]
- We're given that the total perimeter is 80 feet.
3. Set Up the Equation:
- Substituting the length and width into the perimeter formula:
[tex]\[
2l + 2w = 80
\][/tex]
- Replace [tex]\( l \)[/tex] with [tex]\( 3w \)[/tex]:
[tex]\[
2(3w) + 2w = 80
\][/tex]
- Simplify the equation:
[tex]\[
6w + 2w = 80
\][/tex]
[tex]\[
8w = 80
\][/tex]
4. Solve for [tex]\( w \)[/tex]:
- Divide both sides of the equation by 8 to isolate [tex]\( w \)[/tex]:
[tex]\[
w = \frac{80}{8}
\][/tex]
[tex]\[
w = 10
\][/tex]
5. Find the Length:
- Recall that the length [tex]\( l \)[/tex] is three times the width:
[tex]\[
l = 3w
\][/tex]
- Substitute the value of [tex]\( w \)[/tex] into this equation:
[tex]\[
l = 3 \times 10
\][/tex]
[tex]\[
l = 30
\][/tex]
Therefore, the largest possible length of the fence is [tex]\( \boxed{30} \)[/tex] feet.
