A rancher wants to build a rectangular corral for his horses. He Also wants to split the corral into two identical rectangular areas by building a fence through the middle.

Question

If the rancher has 1800 feet of fence, what dimensions should he use for x and y that will maximize the area in the corrals?

Answer 1

Use this illustration to the fence:

x x

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y| y| |y

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x x

Therefore:

Area = A = 2xy

Length of the fence = L = 4x+3y =1800

For the area:

y= A/2x

Replace y with A/2x for the second equation:

4x+3(A/2x) = 1800

3A/2x = 1800 - 4x

3A = 3600x - 8x²

A = 1200x -(8/3)x²

dA = 1200 - (16/3)x

To get the local maximum, dA = 0

0 =1200 - (16/3)x

(16/3)x =1200

x = 225 ft.

To get y, use the second equation above:

4(225)+3y = 1800

900 + 3y = 1800

3y = 900

y = 300 ft.

The maximum area you can have is A=2(225)(300)= 135000 sq. ft.