Question

A solid right circular cone of diameter 14cm and height 8cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm,find the internal diameter of the sphere.

Answer 1

The internal diameter of the sphere is 6 cm.

Explanation:

Given that, a solid right circular cone of diameter 14 cm and height 8 cm.

The radius of the cone is
`=(diameter)/(2)`

`=(14)/(2) \ cm`

= 7 cm.

The volume of the cone is
`= \frac13 \pi r^2 h`

`=(\frac1 3* \pi * 7^2* 8) \ cm^3`

`=(392)/(3)\pi \ cm^3`

Let the internal radius of the sphere be r.

The external diameter of the sphere is = 10 cm

The external radius of the sphere is(R) = 5 cm

The volume of the sphere is
`= \frac43 \pi(R^3-r^3)`

`=\frac43 \pi (5^3-r^3) \ cm^3`

The sphere is formed by the solid right circular cone.

∴The volume of the sphere = The volume of the cone

According to the problem,

`\frac43 \pi (5^3-r^3) =(392)/(3)\pi`

`\Rightarrow 4(5^3-r^3)= 392`

`\Rightarrow 5^3-r^3=(392)/(4)`

`\Rightarrow 5^3-r^3=98`

`\Rightarrow -r^3=98-125`

`\Rightarrow-r^3= -27`

⇒r= 3

The internal radius of the sphere is = 3 cm.

The internal diameter of the given sphere is = (2×3) cm =6 cm.