Height of stacked balls

Question

if there are two balls (size 40 mm and 30mm) stacked in a tube of internal diameter of 50mm, how could you find the height from the bottom of the pipe to the top of the highest stacked ball?

Answer 1

Let $A$ and $B$ denote the midpoints of the two balls. Their radii are $15$ and $20$, so the distance between $A$ and $B$ is $15+20=35$. The horizontal distance between $A$ and $B$ is $50-15-20=15$, hence by Pythagoras theorem the vertical distance between $A$ and $B$ is $$\sqrt{35^2-15^2}=10\sqrt{10}.$$ Then the total height of the stacked balls is $20+10\sqrt{10}+15=35+10\sqrt{10}\approx66.62277660\dots$