A farmer wants to build a rectangular garden. He plans to use a side of the straight river for one side of the garden, so he will not place fencing along this side of the garden. He has 92 yards of fencing material. What is the maximum area that will be enclosed?

Question

26-05-2018
Mathematics

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A farmer wants to build a rectangular garden. He plans to use a side of the straight river for one side of the garden, so he will not place fencing along this side of the garden. He has 92 yards of fencing material. What is the maximum area that will be enclosed?

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tramserran tramserran · Answer 1 · 2018-05-26T14:48:03+02:00

Perimeter = Length(L) + 2 · Width(W)
   92 = L + 2W
92 - 2W = L

Area(A) = Length(L) · Width(W)
       = (92 - 2W) · (W)
       = 92W - 2W²

To find maximum area, calculate the derivative and set it equal to zero.
dA/dW = 92 - 4W
   0 = 92 - 4W
    4W = 92
   W = 23
Now, solve for Length: L = 92 - 2W
       = 92 - 2(23)
       = 92 - 46
         = 46

Lastly, calculate the Area: A = L · W
   = 46 · 46
   = 2116

Answer: The maximum area is 2116 yd²