Question

A farmer wants to make a rectangular pasture with 80,000 square feet. If the pasture lies along a river and he fences the remaining three sides, what dimension should he use to minimize the amount of fence needed?

Answer 1

`200 xx 400` feet

Explanation:

If there were no river and he wanted to fence double that area then he would require a square of side `sqrt(160000) = 400` feet, requiring a total of `4 xx 400 = 1600` feet of fencing.

For the rectangular pasture, imagine the river running through the middle, halving the area and halving the fencing.

So he will require `800` feet of fencing to enclose `80000` square feet, resulting in a pasture `200 xx 400` feet

We can also find/prove this using a little calculus...

Suppose the side of the rectangle parallel to the river is of length `t` feet where `t > 0`

Then the other sides are of length `80000/t`, resulting in a total length used:

`t + 160000/t`

Differentiating this with respect to `t` we get:

`1-160000/t^2`

which is `0` when `t = sqrt(160000) = 400`.

Hence the only (positive) turning point is when `t=400`, which we can quickly verify is a minimum.