You are enclosing a rectangular portion of your lawn with a limited amount of fencing. You want to maximize the amount of area enclosed by this amount of fencing. Find the ratio of length to width of the rectangle with maximum area. Describe the rectangle. How do I solve this if there are no numbers?

The maximum amount of area that can be enclosed by a set length of fence is always a perfect square. So the ratio of length to width should be 1:1.

This may seem silly but consider if you had only 8 feet of fencing. The best possible option would be to have 4 sides of 2 feet a piece for a total of 4 square feet.

2 * 2 = 4

Any other option gets increasingly smaller. For instance, if we had two sides of 3 and two sides of 1, that only gives us 3 square feet.

3*1 = 3

Then if we make two sides 3.5 feet and another. 5 feet it gets even smaller.

3.5*.5 = 1.75

This shows us a trend and allows us to know that a square is the maximum area.