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

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.

Answer 2

Answer:

$A = 64\,ft^{2}$

Step-by-step 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:

$A = 64\,ft^{2}$