A rectangular garden of area 832 square feet is to be surrounded on three sides by a brick wall costing $8 per foot and on one side by a fence costing $5 per foot. find the dimensions of the garden such that the cost of the materials is minimized.

Nice problem!

Area of a rectangle = length * width, or length = Area / width

Let the Width be W

then the length is 832/W

Assume the side W is a fence, and the rest brick.

Total cost, C (W) = $5*W + $8W + 2*$8 * (832/W)

Simplifying, C (W) = 13W+13312/W

To find the minimum cost, we differentiate the cost with respect to W, and equate the derivative C' (W) to zero.

then C' (W) = 13-13312/W^2=0

Rewrite C' (W) as (13W^2-13312) / (W^2) = 0, we solve for W and get

W=sqrt (13312/13) = 32

Therefore length, L=832/32=26

Check L*W=26*32=832 ok [ note L>W, because this costs less $]

Check L=26 and W=32 is the minimum (as opposed to maximum),

we calculate C" (W) = 26624/W^3 > 0 which means that W=26 is a minimum for C (W).

So the dimensions of the garden are 26' x 32', with fence on the W=32' dimension.

Just by curiosity, total cost = C (32) = $832, and average cost = $7.17/'

all sound reasonable.