A circle has a constant diameter. The "largest circle" is the one with the largest diameter. Using a rectangular piece of carpet that is 10 ydx x 7 yds, the largest diameter is 7 yds.
"square feet" is an area.
The area of a circle is Πr2.
Now, since Π does not have an exact decimal representation, we approximate it using 3.14. And, since 1/00th of a foot is difficult enough to measure when cutting carpet, that is a "close enough" answer.
However, this problem uses a "magic number" (that is 7) because a less accurate approximation of Π is 22/7 (note: use this concept when accuracy of answer allows it).
A = (22/7)*(7)2
A = 22*7
A = 154 ft2
Note that if you used Π=3.14, you would get A=153.86 ft2. With carpet, you would have a very difficult time cutting 7/50ths of a square foot.