Question

A park ranger has 32 feet of fencing to fence four sides of a rectangular recycling site. What is the greatest area of recycling site that the ranger can fence? Explain how you know.

Answer 1

`A = 64\,ft^(2)`

Explanation:

The formulas for the perimeter and area of the rectangle are, respectively:

`2\cdot x + 2\cdot y = 32\,ft`

`A = x\cdot y`

After some algebraic handling, the formula for area is simplified into the following form:

`A = x\cdot (16\cdot ft - x)`

`A = 16\cdot x-x^(2)`

The maximum area is found by means of First and Second Derivative Tests:

`A' = 16 - 2\cdot x`

`A'' = -2`

According to the second derivative, the critical point leads invariantly to an absolute maximum. The value of the critical point is:

`16-2\cdot x = 0`

`x = 8\,ft`

The length of the other side is:

`y = 8\,ft`

The maximum area of the recycling site is:

`A = 64\,ft^(2)`

Answer 2

The greatest area he can fence is 64 ft².

In order to maximize area and minimize perimeter, we use dimensions that are as close to equivalent as possible. 32 feet of fence for 4 sides gives us 8 feet of fence per side. We would have a square whose side length is 8; the area would be 8*8 = 64.