A homeowner plans to enclose a 200 square foot rectangular playground in his garden, with one side along the boundary of his pro
perty. 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.
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
