A fence must be built in a large field to enclose a rectangular area of 43,000ft

Question

A fence must be built in a large field to enclose a rectangular area of 43,000ft^2. The east side of the field has a river, so no fence is needed along that side. Material for the fence costs $43 per foot on the west side and $37 per foot on the north and south sides. Determine a cost function C(x) for building the fence if x represends the length of the west side. Find the minimum cost of building the fence, and the dimensions of the field which produces a minimum cost.  

PLEASE EXPLAIN AND SHOW WORK

Explanation / Answer

If the length is x, as the area is 43000, the width is 43000/x

Then, the fence has a west side of length x and a north and south side, each of width 43000/x.

Then, the fence costs 43 x + 2 * 37 * 43000/x = 43x + 3182000/x

We minimize by finding when the first derivative is 0.

43 - 3182000/x^2 = 0

43x^2 = 3182000

x^2 = 74000

x = 20 sqrt(185) = 272.029410174709

The North and South sides are 43000/x = 430/37 * sqrt(185) = 158.071143750169

The cost is 43 * 20 sqrt(185) + 3182000/(20 sqrt(185)) = 1720 sqrt(185) = $23,394.53

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