# Math Help - Cone

1. ## Cone

How high should you fill a cone with height 10 cm if you are to fill half of it?

2. Hello reiward
Originally Posted by reiward
How high should you fill a cone with height 10 cm if you are to fill half of it?
This is equivalent to saying:
Two cones have the same shape (i.e. the same angle at the vertex), and one has twice the volume of the other. The larger cone has height $10$ cm. What is the height of the smaller?
Well, if two similar solids have corresponding lengths in the ratio $a: b$, then their surface areas are in the ratio $a^2:b^2$ and their volumes are in the ratio $a^3:b^3$. So here, if we suppose the height of the smaller cone is $h$ cm, then the ratio of the heights is $h:10$; and we know that the ratio of their volumes is $1:2$. So:
$h^3:10^3 = 1:2$

$\Rightarrow \frac{h^3}{1000} = \frac{1}{2}$

$\Rightarrow h = \sqrt[3]{500}$
$=7.94$ cm (to 2 d.p.)