Question

If a sphere and a rectangular cylinder having equal radius, the height of the cylinder is double of its radius, then, the ratio of volume of sphere and cylinder is
\[\begin{align}
  & \text{A}.\text{ 1}:\text{2} \\
 & \text{B}.\text{ 2}:\text{1} \\
 & \text{C}.\text{ 2}:\text{3} \\
 & \text{D}.\text{ 3}:\text{2} \\
\end{align}\]

Answer 1

Hint: To solve this question, we need to know the formula of the volume of the cylinder and volume of the sphere. Let cylinder be of radius 'r' and height 'h'. Then, volume of cylinder is \[\pi {{r}^{2}}h\] if sphere is of radius R. Then, volume of sphere is \[4\pi {{R}^{3}}\]

Complete step-by-step solution:
Let the sphere be of radius R.
Given that, the sphere and rectangular cylinder have an equal radius.
Then, the radius of the rectangular cylinder is 'r'.
Let the height of the rectangular cylinder be 'h'.
The sphere and cylinder are of the form as below:
 

Also, given that, the height of the cylinder is double its radius.
\[\Rightarrow \text{h=2r }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)}\]
Volume of sphere is \[\dfrac{4}{3}\pi {{r}^{3}}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (ii)}\]
And volume of rectangular cylinder is \[\pi {{r}^{2}}h\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (iii)}\]
Substituting the value of h = 2r from equation (i) to (iii) we get:
Volume of rectangular cylinder \[\Rightarrow \pi {{r}^{2}}\left( 2r \right)\]
Volume of rectangular cylinder \[\Rightarrow 2\pi {{r}^{3}}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (iv)}\]
The ratio of volume of sphere to cylinder is \[\Rightarrow \dfrac{\text{Volume of sphere}}{\text{Volume of cylinder}}\]
Substituting values from equation (ii) and (iv) we get:
\[\Rightarrow \dfrac{\text{Volume of sphere}}{\text{Volume of cylinder}}=\dfrac{\dfrac{4}{3}\pi {{r}^{3}}}{2\pi {{r}^{3}}}\]
Cancelling ${{r}^{3}}$ on both sides, we get:
\[\text{Ratio=}\dfrac{4\times \pi }{3\times 2\pi }\]
Cancelling $\pi $ on both sides, we get:
\[\text{Ratio=}\dfrac{2}{3}\]
Hence, the ratio of volume of sphere to volume of cylinder is 2:3, option C is correct.

Note: The key point in this question is that radius of the sphere and the radius of the cylinder are equal. It is given in the question that, the radius of the cylinder and sphere is the same, so, it is important to assume the same variable r so that cancellation becomes easy. Assuming two different variables will make the question tough and tricky to solve.