A homeowner plans to enclose a 200 square foot rectangular playground in his garden, with one side along the boundary of his property. His neighbor will pay for one third of the cost of materials on that side. Find the dimensions of the playground that will minimize the homeowner's total cost for materials.

Question

Mathematics

Emerson Whitaker

18 May, 05:45

A homeowner plans to enclose a 200 square foot rectangular playground in his garden, with one side along the boundary of his property. His neighbor will pay for one third of the cost of materials on that side. Find the dimensions of the playground that will minimize the homeowner's total cost for materials.

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Answers (1)

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Charlie Weaver · Answer 1

The dimensions are 50 and 100 square foot

Step-by-step explanation:

Let x = length of fenced side parallel to the side that borders the playground

y = length of each of the other two fenced sides

Then, x + 2y = 200

x = 200-2y

The Area = xy = y (200-2y)

The dimensions of the playground that will minimize the homeowner's total cost for materials when the area of the playground is maximum. He can cover more area but with the same cost.

The graph of the area function is a parabola opening downward.

The maximum area occurs when y = - 200/[2 (-2) ] = 50

=> x = 100

So the dimensions are 50 and 100 square foot