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Question

Sam wishes to enclose an 800 ft^2 rectangular community garden. She’ll enclose three of the sides with wooden fence that costs $4.20/foot to install. The fourth side is visible from the road, so on that side she’ll build a decorative stone wall that costs $6.00/foot to install. Use calculus to determine the minimum possible cost to enclose the garden according to Sam’s specifications.

Answer 1

Suppose the sides are x, y

If the sides with the fence are x,x,y length is 2x+y, cost 4.2 (2x+y)

Wall, side y, length y, cost 6y

Total cost C - 4.2 (2x+y) + 6y = 8.4x + 10.2y

xy = 800 because of the area so y = 800/x

C = 8.4x + 8160/x

dC/dx = 8.4 - 8160/x²

dC/dx = 0 when x² = 8160/8.4 = 971.42, giving x = 31.2

x = 31.2 then C = $ 523.62

Technically you should check that C has a minimum when x = 31.2, by showing the second derivative of C is positive when x = 31.2