Answer :
Final answer:
To minimize fencing, Farmer George should choose a square-shaped pasture with sides of 100 meters each, totaling a perimeter of 400 meters.
Explanation:
The dimensions that will yield the least amount of fencing for Farmer George's rectangular pasture can be found by understanding the relationship between area and perimeter of a rectangle. In this case, the area is given as 20,000 square meters.
To minimize fencing, we need to find the dimensions that will satisfy the area requirement with the least amount of perimeter. Let's denote the length of the pasture as L and the width as W.
The perimeter of the rectangle is given by the formula P = 2(L + W). Since no fencing is needed along the barn (considered as one side), the perimeter reduces to P = L + 2W. To minimize the fencing, we aim to minimize P while still meeting the area requirement, which is LW = 20,000 square meters. By substituting, we get 20,000 = L(20,000/L) = 20,000 + 20,000/L^2. To minimize the perimeter, L must equal W, which means the rectangle should be a square. Therefore, the dimensions that require the least amount of fencing are a square with sides of 100 meters each, resulting in a total perimeter of 400 meters.
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