I received the following question in maths today and I don't know how to tackle it.
"The volume of the ice-cream is half the volume of the cone. The cone has a 3cm radius and height of 14cm. What is the depth of the ice-cream?"
I know the volume of the ice cream is $21\pi$, but I don't know how to figure out the height of the ice-cream. When I use the equation I get the following:
$$21\pi = \frac{\pi r^2h}{3}$$
$$63 = r^2h$$
But I have know idea about how to ascertain the height as the radius (and of course the height) is different in the "new" cone.
Anyone know how to do it?
Best Answer
Hint Consider the bottom empty cone with so-far unknown height $h'$ and radius $r'$. We know that is has half the volume of the whole cone, i.e. $$\frac{21 \pi}{2}= \frac{\pi (r')^2 h'}{3}.$$
Also, the bottom cone has the same angle as the whole cone, which means that $\frac{r}{h}=\frac{r'}{h'}$.
Can you use this to solve for $h'$?