1. A sphere is intersected by two parallel planes which are on the same side of its center and are $3 \mathrm{~cm}$ apart. The circles obtained as the intersections of the sphere with the planes have radii of $9 \mathrm{~cm}$ and $12 \mathrm{~cm}$, respectively.

Calculate the volume of the sphere.
Solution. If we orthogonally project the sphere, along with its intersections, onto a plane normal to the intersecting planes, we obtain the image as shown in the right diagram. From the right triangles $OMN$ and $OST$, according to the Pythagorean theorem, we get the equations

$$
R^{2}=12^{2}+x^{2} \text { and } R^{2}=9^{2}+(3+x)^{2} \text {, }
$$

from which we obtain $x=9$ and $R=15 \mathrm{~cm}$. Therefore, $V=\frac{4 R^{3} \pi}{3}=4500 \pi \mathrm{cm}^{3}$.

![](https://cdn.mathpix.com/cropped/2024_06_05_81dca0670dc808162b1cg-6.jpg?height=476&width=503&top_left_y=810&top_left_x=979)