Problem 764

Since the volumes of the two spheres are equal, their radii are also equal, let this be $R$. The distance between the centers of the two spheres is also $R$. The common part consists of two congruent spherical caps. The volume of a spherical cap can be calculated using the following formula:

$$V=\frac{\pi}{3} m^{2}(3 r-m)$$

where $r$ is the radius of the sphere, and $m$ is the height of the spherical cap. In our case,

$$V_{\text{cap}}=\frac{\pi}{3}\left(\frac{R}{2}\right)^{2}\left(3 R-\frac{R}{2}\right)=\frac{5 \pi R^{3}}{24}$$

the common part is twice this volume. The volume of the sphere is $V_{\text{sphere}}=\frac{4 \pi R^{3}}{3}$.

From the ratio of the two volumes:

$$\frac{2 V_{\text{cap}}}{V_{\text{sphere}}}=\frac{\frac{10 \pi R^{3}}{24}}{\frac{4 \pi R^{3}}{3}}=\frac{5}{16}$$

we find that the volume of the common part is $\frac{5}{16}$ of the volume of the sphere.

Gáspár László (Miskolc, Földes F. Gimn., IV. o.t.)

Note. Students in higher grades used their newly acquired knowledge and determined the volumes not by the formula, but by using integral calculus.

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