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.

2 thoughts on “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.”

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

   [tex]V \: = \frac{1}{3} {\pi \: r}^{2} h[/tex]

   Therefore,

   [tex] \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} [/tex]

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

   Reply

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.

   Reply