Question

The height of a cone is 20 cm. A small cone is cut off from the top by a plane parallel to the base. If its volume be 1/125 of the volume of the original cone, determine at what height above the base the section is made.

Answer 1

We have the following situation as shown in the figure

Let VAB be a cone of height h₁ = VO₁ =20cm. Then from the symmetric triangles VO ₁A and VOA₁, we have

`(VO_1)/(VO)=(O_1A)/(OA_1)`

⇒`20/(VO)=(O_1A)/(OA_1)`

It is given that, volume of the cone VA₁O is`1/125`times the volume of the cone VAB. Hence, we have

`1/3piOA_1^2xxVO=1/125xx1/3piO_1A^2xx20`

⇒`((OA_1)/(O_1A))^2xxVO=4/25`

⇒`((VO)/20)^2xxVO=4/25`

⇒`VO^3=(400xx4)/25`

⇒ VO³ = 16 x 4

⇒ VO = 4

Hence, the height at which the section is made is 20 − 4 = 16 cm.