A park ranger has 32 feet of fencing to fence four sides of a rectangular recycling site. What is the greatest area of recycling

The maximum area that can be enclosed is 64 ft². To achieve the largest area while minimizing the perimeter, the dimensions should be as equal as possible. Allocating 32 feet of fencing for four sides gives us 8 feet per side, resulting in a square with a side length of 8; thus, the area equals 8*8 = 64.

The equations representing the perimeter and area of the rectangle are respectively: After manipulating the equations, we derive the area formula in a simplified way. The highest area is calculated using the First and Second Derivative Tests. Based on the second derivative analysis, the critical point consistently indicates an absolute maximum, which is found. The other side's length is computed as follows: The greatest area of the recycling site amounts to: