Task B-4.5. The sphere passes through the vertices of the lower base of the cube and touches all four edges of the upper base. If the length of the edge of the cube is $a$, what is the radius of the sphere?
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_bde1d6b20cdcb90302c4g-21.jpg?height=788&width=702&top_left_y=457&top_left_x=256)

Sketch

(1 point)

From $R^{2}=\frac{a^{2}}{4}+x^{2}$ and $R^{2}=\frac{a^{2}}{2}+y^{2}$, we have

$$
x^{2}-y^{2}=\frac{a^{2}}{4}
$$

Since $x+y=a$, we get $x-y=\frac{a}{4}$. The system

$$
\left\{\begin{array}{l}
x+y=a \\
x-y=\frac{a}{4}
\end{array}\right.
$$

gives $x=\frac{5 a}{8}$ and thus $R=\frac{a}{8} \sqrt{41}$.

(1 point)