Question

Two cones have their base radii in ratio of 5 : 1 and the ratio of their heights as 1 : 5. Find the ratio of their volumes.

Answer 1

Step-by-step explanation:

Given :-

Two cones have their base radii in ratio of 5 : 1 and the ratio of their heights as 1 : 5. Find the ratio of their volumes.

Solution :-

The ratio of height of cone 1 : 5

The ratio of radius of cone is 5 : 1

We know that (WKT), the Volume of the cone

$V \: = \frac{1}{3} {\pi \: r}^{2} h$

Therefore,

$\frac{V1}{V2} = \frac{ \frac{1}{3} {\pi \: r1}^{2}h1 }{ \frac{1}{3} {\pi \: r2}^{2}h 2 } \\ \\ \frac{V1}{V2} = \frac{ {r1}^{2}h1}{ {r2}^{2}h2 } \\ \\ \frac{V1}{V2} = \frac{10 \: \times \: 1}{1 \: \times \: 5} \\ \\ \frac{V1}{V2} = \frac{2}{1}$

Hence, the ratio of the volume of the cone is 2 : 1

Answer 2

The ratio of their heights is 25:64.

Given, the radius of the bases of the cones are in the ratio 4:5

Let us consider the radius of them to be 4x and 5x.

We know,

Volume of a cone is given as (1/3)Пr²h

h is the height of the cone

For, the cone with radius 4x and height h, volume V = (1/3)П(4x)²h = 16Пx²h/3

For, the cone with radius 5x and height h', volume V' = (1/3)П(5x)²h' = 25Пx²h'/3

Given, V/V' = 1/4

⇒[16Пx²h/3]/[25Пx²h'/3] = 1/4

⇒ 16h/25h' = 1/4

⇒ h/h' = 25/64

This is the ratio of their heights.