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Problem

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14 Nov 2021

Given information

The given information is 

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 chambers including the sphere is formed.

To determine: The maximum height of any bubble tower with chambers using mathematical induction.

Step-by-step explanation

Step 1.

Suppose that the maximum height of a bubble tower of bubbles, where the largest bubble has radius is  .

This is certainly true if .

If we now consider a tower of bubbles, then second to bubbles form a tower of bubbles (apart from the first hemisphere), where the bottom bubble of this sub tower has radius (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 and so the height of the whole tower will be

 

so we need to choose to maximize .

 

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