Farmer Jones has 800 meters of fence. She wishes to construct a rectangular pen

Question

Farmer Jones has 800 meters of fence. She wishes to construct a rectangular pen using the river as one side and dividing the pen into three separate rectangular pens as shown. What should the dimensions of the combined pens be to maximize the area enclosed?

Note: While the drawing suggests there might be some double fences or fences along the river, neither of these is the case. Just a sloppy artist at work. No fence is used along the river and no double fences are used.

The total horizontal dimension should be meters.

The vertical dimension should be meters.

The resulting total area is square meters.

Farmer Jones has 800 meters of fence. She wishes to construct a rectangular pen using the river as one side and dividing the pen into three separate rectangular pens as shown. What should the dimensions of the combined pens be to maximize the area enclosed? While the drawing suggests there might be some double fences or fences along the river, neither of these is the case. Just a sloppy artist at work. No fence is used along the river and no double fences are used.

Explanation / Answer

Let the horizontal dimension be x

Let the vertical dimension be y

then total length of fence = x + ( 4* y)

so , x + 4y = 800 m

x = 800 - 4y

Total area = x*y = (800 - 4y)*y = 800y - 4y^2

for the area to be maximum , dA/dy = 0

dA/dy = 800 - 8y =0

y = 100 m

so, x = 800 - (4*100) = 400 m

so The total horizontal dimension should be = x = 400 m

The vertical dimension should be = y = 100 meters.

The resulting total area is = x*y = 400*100 = 40000 square meters.

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