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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.

User Nemith
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2 Answers

1 vote

Answer:


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)

User EJ Campbell
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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.
User Epieters
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