Question

The height of a cone is 30cm .From its topside a small cone is cut by a plane parallel to its base.If the volume of smaller cone is 1/27 of the given cone,then at what height it is cut from its base

Answer 1

See the attachment for detail solution

Answer 2

Answer:

Smaller cone is cut from 20 cm from the base.

Step-by-step explanation:

Given : The height of a cone is 30 cm .From its topside a small cone is cut by a plane parallel to its base. If the volume of smaller cone is 1/27 of the given cone.

To find : What height it is cut from its base?

Solution :

Let h be the height and r be the radius of smaller cone.

Volume of the cone is $V_1=\frac{1}{3}\pi r^2 h$ .....[1]

Volume of the given cone is $V_2=\frac{1}{3}\pi R^2 (30)$.....[2]

From the property of similar triangles,

$\frac{h}{r} =\frac{30}{R}$

$r=\frac{h\times R}{30}$

Substitute in [1],

$V_1=\frac{1}{3}\pi(\frac{h\times R}{30})^2 h$

$V_1=\frac{1}{3}\pi \frac{h^3\times R^2}{30^2}$

We have given, If the volume of smaller cone is 1/27 of the given cone.

$V_1=\frac{1}{27}V_2$

$\frac{1}{3}\pi \frac{h^3\times R^2}{30^2}=\frac{1}{27}(\frac{1}{3}\pi R^2 (30))$

$h^3=\frac{30^3}{27}$

$h^3=1000$

$h=10$

Therefore, Smaller cone is cut from 30-10=20 cm from the base.