# Cones and spheres

A solid metal cone is melted down and turned into spheres. How many spheres can be made?

## Problem

A solid metal cone has radius 6 cm and height 20 cm.

It is melted down to make spheres of diameter 6 cm.

How many spheres can be made?

*This problem is adapted from the World Mathematics Championships*

## Student Solutions

We need to find the volume of the cone and the volume of metal needed for each sphere.

The volume of a cone is given by $\frac{1}{3}\pi r^2h$, where $r$ is the radius and $h$ is the height. So the volume of the cone, in cubic centimetres, is $\frac{1}{3}\pi \times 6^2\times20=240\pi$.

The volume of a sphere is given by $\frac{4}{3}\pi r^3$. The radius of each of the spheres is 3 cm, so the volume of each of the spheres, in cubic centimetres, is $\frac{4}{3}\pi\times3^3=4\times 9\pi=36\pi$.

$240\div36=6.\dot6=6\frac{2}{3}$, so 6 spheres can be made.