Answer:
To maximize the area of a rectangular recycling site given a fixed perimeter, we need to find the dimensions that would yield a rectangle with the largest possible area.
Let's assume the length of the rectangle is L and the width is W. Since we have four sides, we can write the perimeter equation as:
Perimeter = 2L + 2W = 36 feet
We can rearrange this equation to solve for one of the variables in terms of the other. Let's solve for L:
2L = 36 - 2W
L = 18 - W
Now we can express the area (A) of the recycling site in terms of L and W:
A = L × W
A = (18 - W) × W
A = 18W - W²
To find the maximum area, we can take the derivative of the area equation with respect to W and set it equal to zero:
dA/dW = 18 - 2W = 0
Solving for W:
18 - 2W = 0
2W = 18
W = 9
Substituting this value back into the equation for L:
L = 18 - W = 18 - 9 = 9
Therefore, the dimensions that yield the greatest area for the rectangular recycling site are L = 9 feet and W = 9 feet (a square shape). The maximum area of the recycling site that the ranger can fence is:
A = L × W = 9 ft × 9 ft = 81 square feet.
Step-by-step explanation: