Rectangular Fencing Question

Question

A rectangular area of 5000 square feet is to be enclosed using a river as one of the sides as shown. Find the dimensions of the fence that will minimize the fencing materials used. Verify that what you find will result in the minimum fencing used.

Answer 1

Suppose the three fenced sides are w, w, l (with l parallel to the river)

Area = wl = 5000

L = length of fence = 2w + l = 2w + (5000/w)

dL/dw = 2 - (5000/w²)

d²L / dw² = 10000 / w³

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To find max/min L, dL/dw = 0 giving 2w² = 5000, w² = 2500, w = 50

When w = 50, d²L/dw² > 0, showing w=50 is a minimum

wl = 5000 so l = 5000 / 50 = 100

dimensions are width 50, length 100, total fence length 200.