If the height of two cones are in the ratio of 1 : 4 and the radii of their bases are in the ratio 4 : 1, then the ratio of their volumes is:

1. 1 : 2

2. 2 : 3

3. 3 : 4

4. 4 : 1

Let height of cones be 1a and 4a and radius of the cones be 4b and 1b.

Volume of cone = $\dfrac{1}{3}π \text{r}^2 \text{h}$

Volume of 1st cone, V = $\dfrac{1}{3}π (\text{4b})^2 \text{1a} = \dfrac{1}{3}π \times 16\text{b}^2 \times \text{a}$

Volume of 2nd cone, v = $\dfrac{1}{3}π (\text{1b})^2 \text{4a} = \dfrac{1}{3}π \times \text{b}^2 \times 4\text{a}$

$\Rightarrow \dfrac{\text{V}}{\text{v}} = \dfrac{\dfrac{1}{3}π \times 16\text{b}^2 \times \text{a}}{\dfrac{1}{3}π \times \text{b}^2 \times 4\text{a}} \\[1em] = \dfrac{16}{4} \\[1em] = \dfrac{4}{1}.$

= 4 : 1

Hence, option 4 is the correct option.