Question

Tom wants to build a pen in his yard for his dog. He buys 800 ft of fencing. He concludes that he can only build a rectangle with sides of 300 ft and 100 ft.

Answer 1

That gives him an area of 30,000 square feet.

What about a square with sides of 200 feet?

200 * 200 = 40,000 square feet.

Answer 2

2l+2w=800
l+w=400

so he can build an infinite number of such rectangles as long as the length + width = 400

e.g.
1 by 399
2 by 398
3 by 397
...

.1 by 399.9
.01 by 399.99
etc if you allow sides with non-integer values.