ABC Daycare wants to build a fence to enclose a rectangular playground. The area of the playground is 980 square feet. The fence along three of the sides costs $5 per foot and the fence along the fourth side, which will be made of brick, costs $10 per foot. Find the length of the brick fence that will minimize the cost of enclosing the playground. (Round your answer to one decimal place.)

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22-03-2017
Mathematics

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ABC Daycare wants to build a fence to enclose a rectangular playground. The area of the playground is 980 square feet. The fence along three of the sides costs $5 per foot and the fence along the fourth side, which will be made of brick, costs $10 per foot. Find the length of the brick fence that will minimize the cost of enclosing the playground. (Round your answer to one decimal place.)

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nobillionaireNobley nobillionaireNobley · Answer 1 · 2017-03-31T09:20:47+02:00

Let the length of the brick fence be x and the length of the adjacent side of the brick fence y. Then
Area of the rectangular playground is given by xy = 980 square feet.
The cost of enclosing the playgroung is given by C = 10x + 5x + 5(2y) = 15x + 10y
From xy = 980, y = 980/x
Thus, C = 15x + 10(980/x) = 15x + 9800/x
For minimum cost, dC/dx = 0
dC/dx = 15 - 9800/x^2 = 0
9800/x^2 = 15
15x^2 = 9800
x^2 = 9800 / 15 = 653.33
x = sqrt(653.33) = 25.56 = 25.6

Therefore, the length of the brick fence that will minimize the cost of enclosing the playground is 25.6 feet.