Cone

**reiward** · Mar 22nd 2010, 08:33 PM

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

**Grandad** · Mar 23rd 2010, 02:05 AM

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: