Question

A building is in the form of a cylinder surmounted by a hemispherical dome. The base diameter of the dome is equal to $\frac{1}{3}$ of the total height of the building. If the base radius of the cylinder is 6 cm, then the volume of the building is

• A. $1728\pi c{m}^3$
• B. $729\pi c{m}^3$
• C. $1224\pi c{m}^3$
• D. $1024\pi c{m}^3$

Answer 1

The correct option is C $1224\pi c{m}^3$

Let $r$ and $h$ be the base radius and height of the cylinder respectively.
So, the total height of the dome is $(h+r)$.

Diameter of hemispherical dome = $\frac{1}{3}\times$Total height of the building

$\Rightarrow 2r=\frac{1}{3}(h+r)$

$\Rightarrow 2\times 6=\frac{1}{3}(h+6)$ (Given: $r=6$ cm)

$\Rightarrow 36=h+6$

$\Rightarrow h=30$ cm

∴ Volume of building = Volume of cylinder + Volume of hemisphere

$=\pi r^{2}h+\frac{2}{3}\pi r^3$

$=\pi r^2(h+\frac{2r}{3})$

$=\pi \times 6^2(30+\frac{2\times 6}{3})$

$=36\pi \times 34$

$=1224\pi c{m}^3$

Hence, the correct answer is option (c).