High School

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A farmer has 650 feet of fencing. Does he have enough fencing to enclose a rectangular area of half an acre? Show your work.

Note: 1 acre is equal to 43,560 square feet.

Answer :

The farmer has 650 feet of fencing, and half an acre is 21,780 square feet. A square enclosure maximizes the area for a given perimeter, and a square with a 650-foot perimeter can enclose over half an acre. Thus, the farmer does have enough fencing.

To find out if the farmer has enough fencing, we should determine the perimeter of the area that can be enclosed. Since the shape is rectangular, we'll denote the length as L and the width as W, and the area (A) is L imes W. The perimeter (P) would be 2L + 2W. We know that A = 21,780 square feet and P must be less than or equal to 650 feet for the farmer to have enough fencing.

Using the formula A = LW, we can find that L = A / W. Substituting L in the perimeter formula, we get P = 2(A / W) + 2W. Now, we need to examine if there exists a width W such that P \<= 650 when A = 21,780. Without a specific value for L and W, we can use the geometric principle that a square encloses the largest area for a given perimeter, and a square with a perimeter of 650 feet would have sides of 650 / 4 = 162.5 feet.

Therefore, a square enclosure would have an area of 162.5^2 = 26,406.25 square feet, which is more than half an acre. Thus, the farmer does have enough fencing to enclose half an acre.

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