Chapter 4, Problem 26P

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. (The figure shows the case n = 4.) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1 + n .

To determine

To prove: The maximum height of any bubble tower with n chambers is 1+n by using mathematical induction.

Explanation

Given:

A hemispherical bubble is placed on a spherical bubble of radius 1.

A smaller bubble is then placed on first one and this process is continued.

Calculation:

First derive the expression of the radii of the bubble in the hemispherical bubbles in recurring form.

Let Rn denote the radius of the nth bubble placed.

Then distance up to (n + 1)th bubble is given by d(Rn+1)=Rn2Rn+12+Rn+1Rn

Note that Rn2Rn+12 exists since radii are in decreasing order

Put Rn+1=x thus obtain that d(x)=Rn2x2+xRn .

Differentiate d(x) with respect to x,

d(x)=xRn2x2+1

Set d(x)=0 and deduce that

xRn2x2+1=0xRn2x2=12x2=Rn2x=22Rn

As Rn+1=x thus Rn+1=22Rn

Use second principle of mathematical induction to prove the result.

If n=1 then the maximum height is 2 which is nothing but 1+1 .

Thus, the result holds for n=1 .

Induction hypothesis assume that result holds for n bubbles and prove that it also holds for n+1 .

The bubble which at the bottom has its radius equal to 1 as this is given.

Assume the radius of second bubble to be y

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