A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $20/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 88 square feet, find the dimensions of the garden that minimize the cost.
cost = C = 20 L + 10 (2 w + L)
area = w * L = 88 so L = 88/w
so
C = 20* 88/w + 20 w + 10* 88/w = 20 w + 30*88/w
dC/dw = 0 for min = 20 - 2640/w^2
20 w^2 = 2640
w = 11.5
L = 88/11.5 = 7.65