There are five perfectly spherical scoops of ice cream with various radii placed inside a waffle cone. Each scoop of ice cream is in contact with the adjacent scoop of ice cream. Also, each scoop of ice cream comes in contact all around the waffle cone wall. If the radius of the smallest ice cream scoop is 8 and the radius of the largest ice cream scoop is 18, find the volume of the middle ice cream scoop.
My intuition is that the radius should be the geometric mean of 18 and 8 (which is 12), but I don't know why. Is it really the geometric mean and why/why not?
Also, this isn't homework.
Best Answer
Extend the cone to its tip. Consider the following $2$ shapes:
Shape $1$: The cone, from the tip up to the place where it meets the second scoop from the bottom, together with the bottom scoop and the second scoop.
Shape $2$: The cone, from the tip up to the place where it meets the third scoop from the bottom, together with the second scoop from the bottom, and the third scoop from the bottom.
Shapes $1$ and $2$ are similar. Let $r$ be the scaling factor that gets us from the bottom scoop of Shape $1$ to the top scoop of Shape $1$. Then scaling again by the factor $r$ gets us to the top scoop of Shape $2$. And scaling twice more by the factor $r$ gets us to the top scoop in the diagram of the OP.
Thus $\frac{18}{8}=r^4$. It follows that $r^2=\frac{3}{2}$, and therefore the middle scoop has radius $(8)(3/2)$. Now that we have the radius, the volume is easy to write down.
Remark: We could do basically the same argument by drawing a bunch of similar triangles. But that would take longer, and anyway I wanted to give a pure scaling argument.
The argument above shows that the radius is indeed the geometric mean of $12$ and $18$. Precisely the same argument shows that if the bottom scoop has radius $a$ and the top scoop has radius $b$, then the middle scoop has radius $\sqrt{ab}$.
The same argument shows that if the bottom scoop as surface area $A$, and the top scoop has surface are $B$, then the middle scoop has surface area $\sqrt{AB}$.
The same argument shows that if the cone is not right-circular, again the middle scoop has radius $\sqrt{ab}$. The same applies to waffle cones that have the shape of an upside-down pyramid.