Math Question??

Question

I am trying to figure this out and its driving me nuts because it doesn't come out right. OK ... here goes... I just bought a dog exercise pen. It is made up of 8 panels, each 2 feet wide. If I set it up as a square..meaning..4X4X4X4 feet. So thats 2 feet wide by 2 panels = 4 feet square and that equals 16 Square feet. Right?
But if I set the same 8 panels up as a rectangle, its 3 panels on each side = 6 feet and 1 panel on the end = 2 feet how come it comes to 12 square feet instead of 16'? Its the exact same 8 panels, that are each 2 feet wide. Its making me crazy trying to figure out why? Thanks alot.

Answer 1

The square is a special case of a rectangle that can be defined as the parallelogram with the largest area for a set perimeter.

in this problem you have a set perimeter of 16 feet. That never changes.

let x be the width of a rectangle and h be the height of the rectangle.

we know that 2x + 2h = 16 (the perimeter) and the area function is x * h = A

If we define A in terms of x, A = x + 2 * (8 - x) = 16 - x^2

if you graph this parabola you'll see that the area of the rectangle changes as the dimensions change even though the perimeter never changes.

another way to think about this. Get a piece of rope that is 16 feet long and tie it in a loop. if you pull the loop straight so that you have one rope doubled on itself you have zero area enclosed by the rope. if you open the rope just a little you may only have a couple square inches of area but as you expand the rope in the shape of a rectangle you'll find the largest area it can hold (in the form of a rectangle) is when the rope is in the shape of a square. The most area this rope could hold is if it was a circle.

Answer 2

Don't worry, your math looks right. A square is bigger in area than any other rectangle with the same perimeter.
The optimal shape for getting the most area within a fence is a circle, but with right angles, a square is the best you can do.
If you can do an octagon, it will be slightly bigger.