Please help me with this question!!

Question

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $30 per foot, and on the other three sides by a metal fence costing $10 per ft.

If the area of the garden is 148 square feet, find the dimensions of the garden that minimize the cost.

Length of side with bricks (and opposite side)........... ft.

Length of sides perpendicular to the brick wall .......... ft.

Answer 1

Let x = the length of the brick wall side and the length of the fence parallel to the brick wall. Let y be the length of each of the other 2 metal fence sides perpendicular to the brick side.

Cost = 30(brick length) + 10(parallel fence) + 10 (the two perpendicular parts) = 30x + 10x + 10(2y)

= 40x + 20y

Area, A = xy = 148 square feet

y = 148/x

C(x) = 40x + 20(148/x)

To find the minimum, find C'(x), set that to zero and solve for x.

C'(x) = 40-20(148/x²)

40 -20(148/x²) = 0

40 = 20(148/x²)

x² = 148/2 =74

x = ±√74 Use the positive value since this is a length

y = 148/√74 = 2√74

Length of side with bricks (and opposite side) √74 ft.

Length of sides perpendicular to the brick wall 2√74 ft.