Question

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains $41\frac{19}{21}{m}^3$ 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 total height of the building = internal diameter of the dome = 2 r m
$\therefore$ Radius of building ( or dome) = $\frac{2r}{2}=rm$
Height of cylinder = 2r – r = r m
$\therefore$ Volume of the hemispherical dome cylinder = $\frac{2}{3}\pi r^3{m}^3$
$\therefore$ Total volume of the building = Volume of the cylinder + Volume of hemispherical dome
$=(\pi r^3+\frac{2}{3}\pi r^3)m^3=\frac{5}{3}\pi r^3{m}^3$
According to the question,
Volume of the building = volume of the air
$\Rightarrow \frac{5}{3}\pi r^3=41\frac{19}{21}\Rightarrow \frac{5}{3}\pi r^3=\frac{880}{21}\Rightarrow r^3=\frac{880\times 7\times 3}{21\times 22\times 5}=\frac{40\times 21}{21\times 5}=8m^3$
$\Rightarrow r=2m$
$\therefore$ Height of the building = $2r=2\times 2=4m$