Optimization problems

Question

a rectangular field of 1350 ft^2 is to be enclosed with a fence against an existing wall, so that no fence is needed on that side. The material for the fence cost $3/ft. for the two end sides perpendicular to the wall and $4/ft. for the side parallel to the wall. Find the dimensions of the field of which minimized the cost, and the minimum cost.

Answer 1

1350= xy

2x(3)+4y= c = cost minimize cost with what dimensions, x and y??

y = 1350/x

c = 6x +4(1350)/x take the derivative, set = 0

c' = 6 - 5400/x^2 =0

6x^2 = 5400

x^2 = 900

x = 30 feet for the perpendicular side

y = 1350/30 = 45 feet for the parallel side

minimum cost = 6(30) + 4(45) = 180+180 = $360