# Calculus

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing \$60/ft and on the other three sides by a metal fence costing \$50/ft. If the area of the garden is 102 square feet, find the dimensions of the garden that minimize the cost.

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1. let the width be x
and the length y , were the brick wall is along the length.

the xy = 102
y = 102/x

Cost = 50(2x) + 50y + 60y
= 100x + 110y
= 100x + 110(102/x) = 100x + 11220/x
d(cost)/dx = 100 - 11220/x^2
= 0 for a min of cost
11220/x^2 = 100
100x^2 = 11220
x^2 = 11220/100
x = 105.925/10

x = 10.59 ft
they y = 102/x = 9.63 ft

The math tells me that the wall should be build along the shorter side (makes sense)

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