Malcolm wanted to build a dog run with an area of 64. What is the smallest perimeter the dog run could be

If it needs to be a rectangle, then the rectangle with the smallest perimeter for

a given area is the square. He needs 32 feet of fence, and should fence off a

square that's 8 x 8.

But if he's willing to go to the trouble, the perimeter of a circle with the same area

is even less than the square.

A = (pi) (R²)

R = √ (64/pi).

Circumference = (2 pi) (R) = 2 pi √ (64/pi) = √ (256 pi) = 28.359 (rounded).

That's 11.4% less fence to buy, for a circular run.

But on the other hand, what have you got against the dog? One of

the two main purposes of a dog run is to give the dog a place to run.

Minimizing the perimeter also minimizes the distance where he can get

up some speed and run in a straight line ... freeing up his hips, clearing

the cobwebs from his brain, smelling the air, keeping his claws nice and

worn down. With the emotional well-being of the dog in mind, I'd expect

you'd want to give him the maximum possible straight route inside the

run, which, unfortunately, also maximizes the amount of fence that Malcolm

has to provide.

But I digress. The math is done. The question is answered.

This case is closed.