Question

A hollow sphere of internal and external diameters $4cm$ and $8cm$ respectively is melted into a cone of base diameter $8cm$. Find the height of the cone.

Answer 1

Hint: - When we melt one shape into another shape then volume of both the shapes are same (ideal conditions)

Given:
Diameter of the cone is equal to $8cm$.
So the radius ${r_1}$of the cone$= \dfrac{{{\text{diameter}}}}{2} = \dfrac{8}{2} = 4cm$
As we know the volume of the cone is $\dfrac{1}{3}\pi r_1^2h$, where $h$is the height and ${r_1}$is the radius of the cone respectively.
And we know that Volume of hollow sphere of outer radius$\left( R \right)$and inner radius$\left( r \right)$ $= \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right)$
Outer radius$\left( R \right) = 4cm$and inner radius $\left( r \right) = 2cm$
According to given condition
Volume of resulting cone = volume of hollow sphere
$\dfrac{1}{3}\pi r_1^2h = \dfrac{4}{3}\pi \left( {{R^3} - {r^3}} \right) \\\ \Rightarrow r_1^2h = 4\left( {{R^3} - {r^3}} \right) \\\ \Rightarrow {4^2}h = 4\left( {{4^3} - {2^3}} \right) \\\ \Rightarrow 4h = 64 - 8 = 56 \\\ \Rightarrow h = \dfrac{{56}}{4} = 14cm \\\$
Hence, the height of the cone is$14cm$.

Note: -In such types of questions always remember the formula of volume of hollow sphere with inner and outer radius and the formula of volume of cone, then according to given condition both volume are equal then substitute the given values we will get the required value of the height of the cone.