If the volumes of two cones are in the ratio of 1 : 4 and their diameters are in the ratio 4 : 5, then the ratio of their heights is :

1. 1 : 5

2. 5 : 4

3. 5 : 16

4. 25 : 64

Let Volume of cones be V and v respectively.

⇒ V : v = 1 : 4

Diameters of cones be D and d respectively.

⇒ D = 4b and d = 5b

Radius of 1st cone = $\dfrac{\text{diameter}}{2} = \dfrac{4\text{b}}{2}$ = 2b

Radius of 2nd cone = $\dfrac{\text{diameter}}{2} = \dfrac{5\text{b}}{2}$ = 2.5b

Let the height of cones be h and H respectively.

By formula,

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

$\therefore \dfrac{\text{V}}{\text{v}} = \dfrac{\dfrac{1}{3}π (\text{2b})^2 \text{h}}{\dfrac{1}{3}π (\text{2.5b})^2 \text{H}} \\[1em] \Rightarrow \dfrac{1}{4} = \dfrac{\dfrac{1}{3}π \times \text{4b}^2 \text{h}}{\dfrac{1}{3}π \times 6.25\text{b}^2 \text{H}} \\[1em] \Rightarrow \dfrac{1}{4} = \dfrac{4 \text{h}}{6.25\text{H}} \\[1em] \Rightarrow \dfrac{\text{H}}{\text{h}} = \dfrac{4 \times 4}{6.25} \\[1em] \Rightarrow \dfrac{\text{H}}{\text{h}} = \dfrac{16}{6.25} \\[1em] \Rightarrow \dfrac{\text{H}}{\text{h}} = \dfrac{16}{6.25} \\[1em]$

Multipy and divide by 100 on R.H.S

$\Rightarrow \dfrac{\text{H}}{\text{h}} = \dfrac{16 \times 100}{6.25 \times 100} \\[1em] \Rightarrow \dfrac{\text{H}}{\text{h}} = \dfrac{1600}{625} \\[1em] \Rightarrow \dfrac{\text{H}}{\text{h}} = \dfrac{64}{25}$

∴ h : H = 25 : 64

Hence, option 4 is the correct option.