A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $\displaystyle 14$ cm and the total height of the vessel is $\displaystyle 13$ cm . Find the inner surface area and the volume of the vessel.

Radius $\displaystyle =7 \mathrm{~cm}$ Height of cylindrical portion $\displaystyle =13-7=6 \mathrm{~cm}$ Inner surface area of the vessel $\displaystyle =2 \pi \mathrm{r}^{2}+2 \pi \mathrm{rh}$ \[\begin{array}{l} =2 \times \frac{22}{7} \times 7 \times 7+2 \times \frac{22}{7} \times 7 \times 6 \\ =572 \mathrm{~cm}^{2} \end{array} \]Volume of the vessel $\displaystyle =\frac{2}{3} \pi \mathrm{r}^{3}+\pi \mathrm{r}^{2} \mathrm{~h}$ \[\begin{array}{l} =\frac{2}{3} \times \frac{22}{7} \times 7 \times 7 \times 7+\frac{22}{7} \times 7 \times 7 \times 6 \\ =\frac{4928}{3} \text { or } 1642.67 \mathrm{~cm}^{3} \text { approx. } \end{array} \]Therefore, inner surface area and volume of the vessel is $\displaystyle 572 \mathrm{~cm}^{2}$ and $\displaystyle 1642.67 \mathrm{~cm}^{3}$ respectively.