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 $\displaystyle 10$ cm. What is the height of the smaller?
Well, if two similar solids have corresponding lengths in the ratio $\displaystyle a: b$, then their surface areas are in the ratio $\displaystyle a^2:b^2$ and their volumes are in the ratio $\displaystyle a^3:b^3$. So here, if we suppose the height of the smaller cone is $\displaystyle h$ cm, then the ratio of the heights is $\displaystyle h:10$; and we know that the ratio of their volumes is $\displaystyle 1:2$. So:
$\displaystyle h^3:10^3 = 1:2$
$\displaystyle \Rightarrow \frac{h^3}{1000} = \frac{1}{2}$
$\displaystyle \Rightarrow h = \sqrt[3]{500}$
$\displaystyle =7.94$ cm (to 2 d.p.)
Grandad