47. As shown in the figure, in rectangle $A B C D$, $M$ is the midpoint of $A B$, $A N$ is an arc of a circle with center $D$ and radius 3, $K N$ is an arc of a circle with center $C$, and the area of the shaded region is $10-\frac{5 \pi}{2}$. What is the length of the radius of $K N$?
Answer: 1 .

Solution: Let the radius of $K N$ be $r$, then the length of the rectangle is $3+r$, and the width is 3.
$$
S_{\text {shadow }}=(3+r) \times 3-\frac{\pi}{4} r^{2}-\frac{(3+r)}{4}(3-r)=10-\frac{5 \pi}{2} \text {, }
$$

Solving for $r$ gives $r=1$ or $r=13-\pi$ (discard the latter).

