What are the length of the Adjacent sides? Optimization

Question

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $50/ft and on the other three sides by a metal fence costing $40/ft. If the area of the garden is 82 square feet, find the dimensions of the garden that minimize the cost.

Length of side with bricks x=

Length of adjacent side y=

Answer 1

Let the sides be x and y

Area=x.y=82 so y= 82/x (1)

Cost painting C= 50.x+40.(x+2y) (2)

substitute (1) in (2)

C=90.x+80*82/x

find the derivative

C'=90-82*80/x^2

equate C' to zero and solve for x

x^2=80*82/90=72.8889

x=sqrt( 72.8889)=8.5375 ft

y=82/x=9.6047 ft