Cones and spheres
A solid metal cone is melted down and turned into spheres. How many spheres can be made?
Age
14 to 16
Challenge level
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.