Chapter 4, Problem 20P

To determine

The maximum height of any bubble tower with n chambers using mathematical induction.

Explanation of Solution

Given information:

A hemispherical bubble is placed on a spherical bubble of radius 1.A small hemispherical bubble is placed on the first one. This process is continued until n chambers including the sphere is formed

Calculations:

Suppose that the maximum height of a bubble tower of  $n$  bubbles, where the largest bubble has radius 1 is  $1+\sqrt{n}$ This is certainly true if  $n=1$ .

If we now consider a tower of  $n+1$  bubbles, then second to  $n+1$ st bubbles form a tower of  $n$  bubbles (apart from the first hemisphere), where the bottom bubble (of this sub tower) has radius $0<x<1$  (and the height of this sub tower will be as great as possible).

Thus the height of the part of the tower from the centre of the bottom bubble (of the sub tower) to the top will be $x\sqrt{n}$ and so the height of the whole tower will be

  $H(x)=x\sqrt{n}+1+\sqrt{1-x^2}$  

so we need to choose  $x$  to maximize $H(x)$ .

Since  $H\text{'}(x)=\sqrt{n}-\frac{x}{\sqrt{1-x^2}}$ .

We see that  $H(x)$  is maximized when  $x^2=\frac{n}{n+1}$ . This makes the maximum height of the whole tower $1+\sqrt{n+1}$  .

Thus we deduce, by induction, that the maximum height of a tower of  $n$ bubbles is $1+\sqrt{n}$ . This makes the answer  $1+\sqrt{81}=10$