Question

The ratio of volume of a cylinder: volume of cone: volume of hemisphere of same radius and same height is
a) $1:2:3$
b) $2:3:1$
c) $3:1:2$
d) $1:1:1$

Answer 1

Hint: Here first we assume radius and height for all the three shapes. Then put radius and height in the formula of volume of all the three shapes. Then find the ratio. Use formula for the volume of the cylinder,$V=\pi {{r}^{2}}h$ , Volume of the cone,$V=\frac{1}{3}\pi {{r}^{2}}h$ and Volume of the hemisphere,$V=\frac{2}{3}\pi {{r}^{2}}h$

Complete answer:
Let the radius of the base be r and height be h in cylinder, cone and hemisphere.
As we know that height of the hemisphere = radius of the hemisphere$\Rightarrow h=r$.
Volume of the cylinder = $\pi {{r}^{2}}h=\pi {{r}^{2}}r=\pi {{r}^{3}}$.
Volume of the cone = $\frac{1}{3}\pi {{r}^{2}}h=\frac{1}{3}\pi {{r}^{2}}r=\frac{1}{3}\pi {{r}^{3}}$ [taken r=h]
Volume of hemisphere = $\frac{2}{3}\pi {{r}^{3}}$.
So, now the ratio of the volumes of the cylinder, cone and hemisphere will be -
\[\Rightarrow \pi {{r}^{3}}:\frac{1}{3}\pi {{r}^{3}}:\frac{2}{3}\pi {{r}^{3}}\][Common factor $\pi {{r}^{3}}$ is removed from each terms]
\[\Rightarrow 1:\frac{1}{3}:\frac{2}{3}\][Multiplying all the terms with $''3''$ ]
\[=3:1:2\].
Therefore the required answer is - The ratio of volume of a cylinder: volume of cone: volume of hemisphere of same radius and same height is $3:1:2$

Hence, from the given multiple choices, the option C is the correct answer.

Note:: In such types of problems where every term is unknown take the help of the standard general formula, and accordingly follow the given conditions. Also, keep in mind which terms are taken in ratio as $\frac{1}{2}\text{ and }\frac{2}{1}$ make the major difference. So wisely take the ratio by converting the word statement in the proper mathematical form.