5. A sphere $Z$ with radius $R$ has a spherical segment of height $v (v < R)$ removed. A sphere $K$ with radius $\frac{1}{2} v$ is inscribed in this segment. Additionally, eight identical smaller spheres are inscribed in the segment, each touching the sphere $K$. No two of these smaller spheres share an internal point, and each touches exactly two others. Determine the ratio $v: R$.
SOLUTION. Let $S$ be the center of the sphere from which the segment is cut, $O$ the center of the sphere $K$, and $Q$ the center of one of the smaller inscribed spheres with radius $r$. Let the foot of the perpendicular from point $Q$ to the line $S O$ be $P$. Figure 5 represents a cross-section of the figure by the plane $S O Q$. For the marked segments, we have

$$
|Q O|=\frac{v}{2}+r, \quad |P O|=\frac{v}{2}-r, \quad |Q S|=R-r, \quad |P S|=R+r-v
$$

From the right triangles $O Q P$ and $Q S P$, we obtain the equations

$$
\left(\frac{v}{2}+r\right)^{2}-\left(\frac{v}{2}-r\right)^{2}=|Q P|^{2}=(R-r)^{2}-(R+r-v)^{2},
$$

from which

$$
|P Q|^{2}=2 r v, \quad r=\frac{2 R v-v^{2}}{4 R} .
$$

The distance from the center of each smaller sphere to the axis of the segment is $|P Q|$, and the distance between the centers $Q_{1}, Q_{2}$ of two adjacent smaller spheres is $2 r$. Using the cosine rule on the triangle $Q_{1} Q_{2} P$, we get (Figure 6)

$$
\cos \varphi=\frac{2|Q P|^{2}-4 r^{2}}{2|Q P|^{2}}=1-\frac{r}{v}
$$

Since there are eight smaller spheres, $\varphi=45^{\circ}$ and $\cos \varphi=\frac{1}{2} \sqrt{2}$. Expressing $\mathrm{z}$ from the second equality in (1),

it follows from (2) that

$$
\frac{r}{v}=\frac{2 R-v}{4 R}=\frac{1}{2}-\frac{1}{4} \frac{v}{R}
$$

$$
\frac{v}{R}=2\left(1-2 \frac{r}{v}\right)=2(2 \cos \varphi-1)=2(\sqrt{2}-1) .
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_888989f1419c479ab979g-6.jpg?height=304&width=523&top_left_y=1366&top_left_x=346)

Figure 5

![](https://cdn.mathpix.com/cropped/2024_04_17_888989f1419c479ab979g-6.jpg?height=263&width=214&top_left_y=1408&top_left_x=999)

Figure 6

GUIDING PROBLEMS: