question_answer

A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by  \[\frac{32}{9}cm\]. Find the diameter of the cylindrical vessel.  

Answer:

Given, diameter of sphere \[=12\text{ }cm\]

then, radius of sphere \[(r)=\frac{12}{2}=6\,cm\]

Volume of sphere \[=\frac{4}{3}\pi {{r}^{3}}\]

\[=\frac{4}{3}\times \pi \times {{(6)}^{3}}c{{m}^{3}}\]

Now, sphere is completely submerged in water and rise in water in cylindrical vessel is \[3\frac{5}{9}cm.\]

Volume of sphere = Volume of cylindrical vessel.

\[\frac{4}{3}\pi \times {{(6)}^{3}}=\pi {{r}^{2}}\times \frac{32}{9}\]

\[{{r}^{2}}=\frac{4\times 6\times 6\times 6\times 9}{3\times 32}\]

\[r=\sqrt{81}\]

\[r=9\,\,cm\]

\[\therefore \] Diameter of the cylindrical vessel is \[18\text{ }cm\].