You have 160 yards of fencing to enclose a rectangular region. find the maximum area of the rectangular region

I know that for a fixed perimeter, the shape that encloses the greatest area

is a circle, and the rectangle that encloses the greatest area is a square.

Sadly, I don't know how to prove it to you without Calculus.

If you'll take my assertion that the greatest rectangle is a square, and accept

it on faith, then you should use your 160-yd of fence to enclose a square with

40-yd sides. The area inside it is (40 x 40) = 1600 square yards.

Here are some other choices.

Each one has the same perimeter ... 160 yards.

This table kind of suggests to you that a square is the rectangle

with the greatest area. (But it doesn't prove it.)

Length Width Area

35-yd 45-yd 1,575 square yards

30 50 1,500

25 55 1,375

20 60 1,200

15 65 975

10 70 700

5 75 375

3 77 231

2 78 156

1 79 79

2-ft 79-yd 1-ft 52.89

1-ft 79-yd 2-ft 26.56

1-inch 79-yd 35-in 2.19 square yards

With the same 160-yd of fence, you could have squeezed in some more

area by setting the fence down in a circle with circumference = 160-yd.

The area inside the circle would be

Area = (pi) (radius) ² = (circumference) ² / (4 pi) = 2,037.2 square yards

The area of a circle is always (4 / pi) times the area of the square with the

same perimeter. That's about 27.3% more area than the square.

To optimize the area, the shape should be a square. Clearly, with 160 yards of fencing, the square will have side-length 160/4=40 yards. Therefore, the area will be 40*40=1600 yards.