Question

A sphere of diameter $5cm$ is dropped into a cylindrical vessel partly filled with water. The diameter of the base of the vessel is $10cm.$ If the sphere is completely submerged, by how much will the level of water rise?

Answer 1

Diameter of sphere$=5cm$

Radius of sphere,$,r=\frac{5}{2}cm$

The diameter of the base of the vessel$=10cm$

Radius of the base of the vessel$,R=\frac{10}{2}=5cm$

Volume of sphere $=\frac{4}{3}{\mathrm{\pi r}}^3$

Volume of cylindrical vessel$={\mathrm{\pi R}}^2\mathrm{h}$

Volume of sphere $=$ Volume of water raised in cylindrical vessel.

$$\begin{gathered} \Rightarrow \frac{4}{3}{\mathrm{\pi r}}^3={\mathrm{\pi R}}^2\mathrm{h} \\ \Rightarrow \frac{4}{3}\times {\left(\frac{5}{2}\right)}^3=5^2\mathrm{h} \\ \Rightarrow \mathrm{h}=\frac{\left({\displaystyle \frac{4}{3}}\right){\left({\displaystyle \frac{5}{2}}\right)}^3}{5^2} \\ =\frac{5}{6} \end{gathered}$$

The level of water rise $=\frac{5}{6}cm$