11. Coco is making clown hats from a circular piece of cardboard. The circumference of the base of each hat equals its slant height, which in turn is equal to the radius of the piece of cardboard. What is the maximum number of hats that Coco can make from the piece of cardboard?

A 3
B 4
C 5
D 6
Solution: D
Let the radius of the circle be $r$. This must also be the slant height of each hat. If we flatten out a hat we get a sector of a circle in which the circular part of the boundary has length $r$, as shown in the diagram on the left. The circumference of a circle of radius $r$ has length $2 \pi r$. Since $3<\pi<3.5$, it follows that $6 r<2 \pi r<7 r$. So Coco can cut out 6 sectors from the circular piece of card as shown, each of which can be made into a hat. The area of card needed for each hat is $\frac{1}{2} r^{2}$ and the area of the circle is $\pi r^{2}$. As $\pi r^{2}<7\left(\frac{1}{2} r^{2}\right)$, Coco cannot make 7 hats. So the maximum number of hats that Coco can make is 6 .