Question

If the radii and heights of two cylinders are in the ratio $1:2$ and $2:1$ respectively, then the ratio of their volumes is:

• A. $1:2$
• B. $2:1$
• C. $1:4$
• D. $4:1$

Answer 1

The correct option is A $1:2$

Given, the ratio of the radii and heights of two cylinders are $1:2$ and $2:1$ respectively.

Let the radius of the first cylinder be $x,$ then the radius of the second cylinder will be $2x.$
Similarly, let the height of the first cylinder be $2y,$ then the radius of the second cylinder will be $y.$

We know that, $\text{Volume of a cylinder =}\pi \times \text{(radius)}{}^2\times \text{height}$

Step 1: Volume of the first cylinder $=\pi \times {\text{(radius of the first cylinder)}}^2\times \text{height of the first cylinder}$
$=\pi \times x^2\times 2y$
$=2\pi x^{2}y$

Step 2: Volume of the second cylinder $=\pi \times {\text{(radius of the second cylinder)}}^2\times \text{height of the second cylinder}$
$=\pi \times (2x{)}^2\times y$
$=4\pi x^{2}y$

Step 3: Ratio of the volumes
$=\text{Volume of the first cylinder : Volume of the second cylinder}$
$=2\pi x^{2}y:4\pi x^{2}y$
Divide both sides by $2\pi x^{2}y$
$=\frac{2\pi x^{2}y}{2\pi x^{2}y}:\frac{4\pi x^{2}y}{2\pi x^{2}y}$
$=1:2$

Hence, option (a.) is the correct choice.