## Task 28/62

a) Six smaller, equally sized spheres are to be placed inside a hollow sphere with a diameter of $D=2 R$ such that each of them touches the inner surface of the hollow sphere and four of the smaller spheres. How large must the diameter $d=2 r$ of the smaller spheres be chosen?

b) Eight smaller, equally sized spheres are to be placed inside a hollow sphere with a diameter of $D=2 R$ such that each of them touches the inner surface of the hollow sphere and three of the smaller spheres. The diameter $d=2 r$ of the smaller spheres is to be determined.

c) Four identical spheres are to be placed inside the hollow sphere such that each sphere touches every other sphere. How large is their diameter $d=2 r$?

![](https://cdn.mathpix.com/cropped/2024_06_06_1d2be4837fb839c2b31dg-053.jpg?height=494&width=417&top_left_y=2083&top_left_x=291)
a) The centers of the embedded spheres must lie at the endpoints of an octahedron whose side length \( s = 2r = d \). The center of the octahedron coincides with the center of the hollow sphere. If one makes a planar cut through four (any) edges of the octahedron, the following figure is obtained.

The diameter \( D \) is composed of three segments: \( B B' = B M + M M' + M' B' \).

Now, \( M M' \) is the diagonal of a square with side length \( s = d \). Therefore,

\[
D = r + d \sqrt{2} + r = d + d \sqrt{2} = d(1 + \sqrt{2}) \rightarrow d = \frac{D}{\sqrt{2} + 1}
\]

One multiplies the right-hand fraction by \( \sqrt{2} - 1 \) to rationalize the denominator:

\[
d = D(\sqrt{2} - 1) \approx 0.414 D
\]

b) The centers of the embedded spheres lie at the endpoints of a cube whose center coincides with that of the hollow sphere. If one makes a diagonal cut through the cube, the following figure is obtained as the cross-section.

The sides of the rectangle resulting from the cube's diagonal cut are \( d \) and \( d \sqrt{2} \). The diagonals of the rectangle are the space diagonals of the cube and have a length of \( d \sqrt{3} \). For the composition of the diameter \( D \) of the hollow sphere, it holds that:

\[
D = r + d \sqrt{3} + r = d(1 + \sqrt{3}) \rightarrow d = \frac{D}{\sqrt{3} + 1} \approx 0.366 D
\]

![](https://cdn.mathpix.com/cropped/2024_06_06_1d2be4837fb839c2b31dg-054.jpg?height=846&width=1496&top_left_y=248&top_left_x=286)

c) The centers of the four embedded spheres form a tetrahedron with edge length \( d \). The intersection of the four altitudes of the tetrahedron coincides with the center of the hollow sphere. The height of the tetrahedron is \( \frac{d \sqrt{6}}{3} \). The altitudes divide each other in the ratio \( 3:1 \), measured from the vertices. If one makes a symmetry cut through the tetrahedron and the hollow sphere, the following figure is obtained:

From the figure, it is clear that \( O B_1 = O M_1 + M_1 B_1 \), so

\[
R = \frac{3}{4} \cdot \frac{d \sqrt{6}}{3} = \frac{d}{4}(\sqrt{6} + 2) \rightarrow D = \frac{d}{2}(\sqrt{6} + 2) \rightarrow d = D(\sqrt{6} - 2) \approx 0.449 D
\]