 [

A jeweler was commissioned to make a gold ring of width $h$, having the shape of a body bounded by the surface of a sphere with center $O$ and the surface of a cylinder of radius $r$, the axis of which passes through point $O$. The master made such a ring, but chose $r$ too small. How much gold will he have to add if $r$ needs to be increased by $k$ times, while keeping the width $h$ the same?

#
![](https://cdn.mathpix.com/cropped/2024_05_06_6741038a32a996d2ce33g-24.jpg?height=337&width=468&top_left_y=283&top_left_x=1501)

It is not difficult to prove that the volume of the spherical ring - the body obtained by rotating the segment AB around the diameter MN, which does not intersect this segment, - is equal to $\pi / 6 AB^{2} \cdot A_{1}B_{1}$, where $A_{1}B_{1}$ is the projection of the chord AB onto the diameter MN. This formula is given, for example, in the book by J. Hadamard "Elementary Geometry," Vol. II, Ch. IV, §500. The volume of the ring is $\pi / 6 h^{3}$, i.e., it does not depend on the radius $r$, so the master will not need to add any gold.