# A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom.

Question:

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius is 60 cm and its height is 180 cm.

Solution:

Height of the conical part = 120 cm

Base radius of the conical part = 60 cm

$\therefore$ Volume of the conical part $=\frac{\mathbf{1}}{\mathbf{3}} \pi \mathrm{r}^{2} \mathrm{~h}$

$=\frac{\mathbf{1}}{\mathbf{3}} \times \frac{\mathbf{2 2}}{\mathbf{7}} \times 60^{2} \times 120 \mathrm{~cm}^{3}$

Radius of the hemispherical part = 60 cm.

$\therefore$ Volume of the hemispherical part $=\frac{\mathbf{2}}{\mathbf{3}} \pi \mathrm{r}^{3}$

$=\frac{2}{3} \times \frac{22}{7} \times(60)^{3} \mathrm{~cm}^{3}$

$\therefore$ Volume of the solid $=$ [Volume of conical part] $+$ [Volume of hemispherical part]

$=\left[\frac{1}{3} \times \frac{22}{7} \times 60^{2} \times 120\right]+\left[\frac{2}{3} \times \frac{22}{7} \times 60^{3}\right] \mathrm{cm}^{3}$

$=\frac{\mathbf{2}}{\mathbf{3}} \times \frac{\mathbf{2 2}}{\mathbf{7}} \times 60^{2}[60+60] \mathrm{cm}^{3}$

$=\frac{2}{3} \times \frac{22}{7} \times 60 \times 60 \times 120 \mathrm{~cm}^{3}=\frac{6336000}{7} \mathrm{~cm}^{3}$

$\Rightarrow$ Volume of water in the cylinder $=\pi \mathrm{r}^{2} \mathrm{~h}$

$=\frac{\mathbf{2 2}}{\mathbf{7}} \times 60 \times 60 \times 180$

$=\frac{14256000}{7} \mathrm{~cm}^{3}$

$\therefore$ Volume of water left in the cylinder

$=\left[\frac{14256000}{7}-\frac{6336000}{7}\right] \mathrm{cm}^{3}$

$=\frac{7920000}{7} \mathrm{~cm}^{3}$

$=1131428.57142 \mathrm{~cm}^{3}=\frac{\mathbf{1 1 3 1 4 2 8 . 5 7 1 4 2}}{\mathbf{1 0 0 0 0 0 0}} \mathrm{m}^{3}$

$=1.13142857142 \mathrm{~m}^{3}=1.131 \mathrm{~m}^{3}$ (approx).