Question

A right circular cone and a cylinder have a circle of unit radius as base and their heights are equal to the radius itself and a hemisphere has the same radius then their volumes are proportional respectively to-
A.$1:2:3$
B.$3:2:1$
C.$2:1:3$
D.$1:3:2$

Answer 1

Hint: First use the formula of volume of cone which is given as-
Volume of cone=$\dfrac{1}{3}\pi {r^2}h$ where r is radius and h is height, then formula of volume of cylinder which is given as-
Volume of cylinder=$\pi {r^2}h$ where r is radius and h is height and then use the that the volume of hemisphere which is given as-
Volume of hemisphere=$\dfrac{2}{3}\pi {r^3}$where r is radius. Then put the values according to the question and find their ratios by putting their volumes in this-
Volume of cone: Volume of cylinder: Volume of hemisphere

Complete step-by-step answer:
Given, the right circular cone has unit base r and its height is equal to the radius which means h=r.
The cylinder also has the same height as its radius and its radius is r then its height h= r.
The hemisphere has the same radius as the cone and cylinder r.
We have to find the ratio of their volumes.
So first we will find their volumes.

We know that the volume of cone is given as-
Volume of cone=$\dfrac{1}{3}\pi {r^2}h$
Then according to question the height h=r then volume becomes-
Volume of cone=$\dfrac{1}{3}\pi {r^2} \times r = \dfrac{1}{3}\pi {r^3}$ -- (i)
Now we know that volume of cylinder is given as-
Volume of cylinder=$\pi {r^2}h$
But according to question h=r then we get,
Volume of cylinder=$\pi {r^2} \times r = \pi {r^3}$ --- (ii)
Also we know that the volume of hemisphere is given as-
Volume of hemisphere=$\dfrac{2}{3}\pi {r^3}$ -- (iii)
Now we can write the ratio of their volumes as-
Volume of cone: Volume of cylinder: Volume of hemisphere
On putting values we get,
$ \Rightarrow \dfrac{1}{3}\pi {r^3}:\pi {r^3}:\dfrac{2}{3}\pi {r^3}$
On dividing the three volumes by $\pi {r^3}$, we get-
$ \Rightarrow \dfrac{1}{3}\dfrac{{\pi {r^3}}}{{\pi {r^3}}}:\dfrac{{\pi {r^3}}}{{\pi {r^3}}}:\dfrac{2}{3}\dfrac{{\pi {r^3}}}{{\pi {r^3}}}$
On division we get,
$ \Rightarrow \dfrac{1}{3}:1:\dfrac{2}{3}$
On multiplying by $3$ on each term we get,
$ \Rightarrow \dfrac{3}{3}:3:\dfrac{{2 \times 3}}{3}$
On solving we get,
$ \Rightarrow 1:3:2$
Hence the correct answer is D.

Note: There is a difference between hemisphere and sphere. Hemisphere is the half part of sphere or we can say the sphere is divided into two equal parts called hemisphere and we know that the volume of sphere is given as-
Volume of sphere=$\dfrac{4}{3}\pi {r^3}$ where r is the radius of the sphere. We just divide the formula of sphere by $2$ to get the formula of hemisphere-
Volume of hemisphere=$\dfrac{1}{2} \times \dfrac{4}{3}\pi {r^3}$
On dividing we get,
Volume of hemisphere=$\dfrac{2}{3}\pi {r^3}$