The ratio between the heights of two solid cones is 2 : 3 and ratio between their radii is 9 : 8. The ratio between their volumes is :

1. 27 : 32

2. 32 : 27

3. 3 : 2

4. 2 : 3

Given,

Ratio between the heights of two solid cones is 2 : 3.

Let height of first (h₁) and second cone (h₂) be 2a and 3a.

Ratio between the radii of two solid cones is 9 : 8.

Let radii of first (r₁) and second cone (r₂) be 9x and 8x.

$\dfrac{\text{Vol. of 1st cone}}{\text{Vol. of 2nd cone}} = \dfrac{\dfrac{1}{3}πr$1^2h1}{\dfrac{1}{3}πr2^2h2} \\[1em] = \dfrac{r1^2h1}{r2^2h2} \\[1em] = \dfrac{(9x)^2 \times 2a}{(8x)^2 \times 3a} \\[1em] = \dfrac{81x^2 \times 2a}{64x^2 \times 3a} \\[1em] = \dfrac{162x^2a}{192x^2a} \\[1em] = \dfrac{27}{32} \\[1em] = 27 : 32.

Hence, Option 1 is the correct option.