Question

Find the dimensions of the rectangular corral split into two pens (sharing a fence) of the same size producing the greatest enclosed area given 630 630 feet of fencing.

Answer 1

             L
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|                             |    w
|                             |
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|                             |
|                             |  w
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Area = l* 2w

fence length: 3l + 4w = 630 =>  3l = 630 - 4w => l = 630 / 3 - (4w/3)

=> l = 210 -4w/3


=> area = (210 - 4w/3) * (2w) = 420w - 8(w^2)/3

The maximum or minimum of a parabola of the form ax^2 + bx + c is at x = -b/2a


So, the maximum of - 8(w^2) / 3 + 420 w is at w = - 420 / ( - 2*8/3) = 78.75

and l = 210 - 4(78.75)/3 = 105.

Answer: l = 105 feet and 2* w = 157.5 feet