Question

A building is in the form of a cylinder surmounted by a hemi-spherical vaulted dome and contains $41\frac{19}{21}{\mathrm{m}}^3$ of air. If the internal diameter of dome is equal to its total height above the floor , find the height of the building ?

Answer 1

let the total height of the building be H m.

let the radius of the base be r m. Therefore the radius of the hemispherical dome is r m.

Now given that internal diameter = total height
$\Rightarrow 2r=H$

Total height of the building = height of the cylinder +radius of the dome
⇒ H = h + r
⇒ 2r = h + r
⇒ r = h

Volume of the air inside the building = volume of the cylinder+ volume of the hemisphere
$$\begin{gathered} \Rightarrow 41\frac{19}{21}={\mathrm{\pi r}}^2\mathrm{h}+\frac{2}{3}{\mathrm{\pi r}}^3 \\ \Rightarrow \frac{880}{21}={\mathrm{\pi h}}^2\mathrm{h}+\frac{2}{3}{\mathrm{\pi h}}^3 \\ \Rightarrow \frac{880}{21}={\mathrm{\pi h}}^3\left(1+\frac{2}{3}\right) \\ \Rightarrow \frac{880}{21}={\mathrm{\pi h}}^3\left(\frac{5}{3}\right) \\ \Rightarrow \mathrm{h}=2\mathrm{m} \end{gathered}$$

Hence, height of the building H = 2 × 2 = 4m