13.244. Two balls are placed in a cylindrical jar with a diameter of 22 cm (Fig. 13.4). If 5 liters of water are poured into the jar, will both balls, with diameters of 10 and 14 cm, be completely covered by the water?
## Solution.

Consider the plane $\beta$, which is perpendicular to the base of the cylinder and contains the line on which the centers of the spheres $O_{1}$ and $O_{2}$ lie. $\Delta O_{1} B O_{2}$ is a right triangle, $O_{1} O_{2}=5+7=12 \mathrm{~cm}$,

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1168.jpg?height=589&width=504&top_left_y=1197&top_left_x=723)

Fig. 13.4 $O_{2} B=22-(5+7)=10 \mathrm{~cm}$.

By the Pythagorean theorem: $B O_{1}=\sqrt{12^{2}-10^{2}}=2 \sqrt{11} \mathrm{~cm}$.

Then $h=7+2 \sqrt{11}+5=12+2 \sqrt{11}$ cm. The volume of water needed to cover the spheres: $V=h \cdot \pi \cdot \frac{22^{2}}{4}-\frac{4}{3} \pi\left(7^{3}+5^{3}\right) \approx 5123 \mathrm{~cm}^{3}$.

Since $5123>5000\left(\right.$ cm $\left.^{3}\right)$, the spheres will not be completely covered by water.

Answer: They will be slightly uncovered.