Question

A gardener builds a rectangular fence around a garden using at most 56 feet of fencing. The length of the fence is four feet longer than the widthWhich inequality represents the perimeter of the fence, and what is the largest measure possible for the length?

Answer 1

The inequality representing the perimeter of the fence is 2w + 2(w + 4) ≤ 56, with w being the width. The largest possible length of the fence is 16 feet.

Step-by-step explanation:

The question refers to finding an inequality that represents the perimeter of a rectangular fence with certain dimensions. Given that the length of the fence is four feet longer than its width and using at most 56 feet of fencing:

Let w represent the width in feet. The length, which is four feet longer than the width, would then be w + 4 feet.

The perimeter of a rectangle is obtained by adding the lengths of all four sides together, so we have:

2(w) + 2(w + 4) ≤ 56

Simplify the inequality:

2w + 2w + 8 ≤ 56

4w ≤ 48

w ≤ 12

Substitute the largest possible width (w = 12) into the length expression to find the largest possible length:

Largest length = w + 4 = 12 + 4 = 16 feet

The inequality that represents the perimeter condition is 2w + 2(w + 4) ≤ 56, and the largest possible length of the fence is 16 feet.

Answer 2

We know that

• The gardener used at most 56 feet of fencing.

,

• The length of the fence is four feet longer than the width.

Remember that the perimeter of a rectangle is defined by

`P=2(w+l)`

Now, let's use the given information to express as inequality.

`2(w+l)\leq56`

However, we have to use another expression that relates the width and length.

`l=w+4`

Since the length is 4 units longer than the width. We replace this last expression in the inequality.

`\begin{gathered} 2(w+w+4)\leq56 \\ 2(2w+4)\leq56 \\ 2w+4\leq(56)/(2) \\ 2w+4\leq28 \\ 2w\leq28-4 \\ 2w\leq24 \\ w\leq(24)/(2) \\ w\leq12 \end{gathered}`

The largest width possible is 12 feet.

Now, we look for the length.

`\begin{gathered} 2(12+l)\leq56 \\ 24+2l\leq56 \\ 2l\leq56-24 \\ 2l\leq32 \\ l\leq(32)/(2) \\ l\leq16 \end{gathered}`

Therefore, the largest measure possible for the length is 16 feet.