# Optimization Calculus

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A three sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region. There is 96 ft of fencing available. Find the maximum enclosed area and the dimensions of the corresponding enclosure.

I drew a picture of it and I got the area = x*y. I used x as the two widths and y as the length on the picture.

• Optimization Calculus -

so, now go for it. You know that

2x+y = 96, so y = 96-2x

the area is xy = 2x(96-2x) = 192x - 4x^2

That's just a parabola; its vertex gives the maximal area.

• Optimization Calculus -

since there are usually four sides to a square(which is the optimal shape), you would do 96/4 to figure out what one side would be. Then since one side is missing, you'd add that one side to the other side. So s= side, 2s to find the other side length, which would then be 48. So the dimensions would just be, 24 by 24 by 48 to get the maximum area.

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