A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $40/ft and on the other three sides by a metal fence costing $20/ft. If the area of the garden is

Question

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $40/ft and on the other three sides by a metal fence costing $20/ft. If the area of the garden is 22 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

This is an optimization problem, we have to define the constraints, and a cost equation in terms of one variable, then differentiate it and set it to 0 to find the minimum cost.

Lets assume the garden has length L and width W and one Length has the brick wall.

L*W = 22 (area constraint)

C = 40*L + 20*L + 20*W + 20*W = 60*L + 40*W

We need the equation in terms of just one variable, lets use L. We can use the area constraint to do this

W = 22/L (plug this into the C equation)

C(L) = 60*L + 40*(22/L) = 60*L + 880/L

C(L) = 60*L + 880/L

now take the derivative and get

C'(L) = 60 - 880/L^2

now set that derivative = 0 to find the key L value where the min/max occurs (min in this case)

0 = 60 - 880/L^2

Solve for L

L = ± (2 sqrt(11/3))

We only need the positive answer as negative length makes no sense here

L = 3.83 (approximate)

We can now solve for W as well

W = 22/3.83 = 5.74

So the minimum cost occurs at L = 3.83 and W = 5.74

Try graphing the original Cost function C(L) to see how the cost curves down then hits a min at L = 3.83 and goes back up again. Please let me know if you have other questions.