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.)

Question

Mathematics

Sheldon Barker

9 June, 16:36

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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Bongo · Answer 1

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.