A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical…

Let r is the radius of hemisphere and cylinder and h is the height of cylinder \[h=2 r \] Volume of air in room \(\displaystyle =\frac{2}{3} \pi \mathrm{r}^{3}+\pi \mathrm{r}^{2} \mathrm{~h}\) \[\begin{aligned} \frac{1408}{21} & =\frac{2}{3} \pi r^{3}+\pi r^{2}(2 r) \\ \frac{1408}{21} & =\frac{8}{3} \times \frac{22}{7} \times r^{3} \\ r^{3} & =8 \\ \therefore \quad r & =2 \mathrm{~m} \\ \text { and } h & =4 \mathrm{~m} \end{aligned} \]