Question

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:

Options

• 27 : 32

• 32 : 27

• 3 : 2

• 2 : 3

Answer 1

27 : 32

Explanation:

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.

\[\frac{\text{Vol. of 1st cone}}{\text{Vol. of 2nd cone}} = \frac{\frac{1}{3}\, \pi r_1^2 h_1}{\frac{1}{3}\, \pi r_2^2 h_2}\]

\[{} = \frac{r_1^2 h_1}{r_2^2 h_2}\]

\[{} = \frac{(9x)^2 \times 2a}{(8x)^2 \times 3a}\]

\[{} = \frac{81x^2 \times 2a}{64x^2 \times 3a}\]

\[{} = \frac{162x^2 a}{192x^2 a}\]

\[{} = \frac{27}{32}\]

= 27 : 32