Question

What is the volume of a small cone if:

Two cones are similar. The lateral area of the small cone is equal to 60pi cm squared. The radius and the slant height of the large cone measure respectively 9cm and 15cm. What is the volume of the small cone?

Answer 1

Volume of smaller cone =301.44

Explanation:

First let angle of cone be
`\alpha`.

then
`sin(\alpha )` = 3/5 (For bigger triangle)

Same cone angle is for smaller one because they are similar.

`cos(\alpha )` = 4/5

Now for smaller cone radius be R

3/5 =R/Slant height

Slant height =5/3R

Lateral surface area =
`(1)/(2)R^(2)\beta` [R = slant height]

Given lateral surface area =60
`\pi` =
`(1)/(2)((5R)/(3) )^(2) \beta` ...................................1

`2\pi R=(5R)/(3)\beta` ................................................2

`\beta=(6\pi )/(5)`

Put angle beta in equation 1 and find the value of R

R =6 cm

Height =
`(4R)/(3)`

Thus volume =
`(1)/(3)\pi6^(2)*8`

=301.44