A fence is to be built to enclose a rectangular area of 220 square feet. The fence along three sides is to be

A fence is to be built to enclose a rectangular area of 220 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 16 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Dimensions: x

First we imagine a rectangle with long side x and small side y. Obviously there are two sides of x opposite each other and similarly for y. So clearly we want to minimise the cost for y as one is $5 per foot and the other $16 per foot.

Our first equation comes from the area: x * y = 220.

Our second comes from the cost: C = 5x + 5x + 5y + 16y, where C is the cost for the whole enclosure and this is simplified as C = 10x + 21y.

Now we get rid of the x in our C cost equation using equation one which yields: x = 220 / y, so our cost equation becomes: C = 2200/y + 21y. Now we want to minimise this using calculus.

dC/dy = -2200/y^2 + 21 and now setting C = 0 to find our minimum point we get:

-2200/y^2 + 21 = 0 => 21y^2 = 2200 => y ≈ 10.24 feet. I verified that this was in fact a minimum at 10.24 as the second derivative at 10.24 is positive and hence a minimum (It's easy to check yourself by looking at the second derivative).

Now subbing this into equation one yields: x ≈ 21.48 and thus our minimum cost for the enclosure is: 10(21.48) + 21(10.24) ≈ $430.

So in brief, let the longer side of your rectangle be 21.48 feet and the smaller side 10.24 feet and you will build the most economical enclosure.