BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Solutions

Chapter
Section
BuyFindarrow_forward

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 .

Chapter 4, Problem 26P, A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble

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

This gives the relation between successive radii in term of its adjacent radius.

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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started
Sect-4.1 P-11ESect-4.1 P-12ESect-4.1 P-13ESect-4.1 P-14ESect-4.1 P-15ESect-4.1 P-16ESect-4.1 P-17ESect-4.1 P-18ESect-4.1 P-19ESect-4.1 P-20ESect-4.1 P-21ESect-4.1 P-22ESect-4.1 P-23ESect-4.1 P-24ESect-4.1 P-25ESect-4.1 P-26ESect-4.1 P-27ESect-4.1 P-28ESect-4.1 P-29ESect-4.1 P-30ESect-4.1 P-31ESect-4.1 P-32ESect-4.1 P-33ESect-4.1 P-34ESect-4.1 P-35ESect-4.1 P-36ESect-4.1 P-37ESect-4.1 P-38ESect-4.1 P-39ESect-4.1 P-40ESect-4.1 P-41ESect-4.1 P-42ESect-4.1 P-43ESect-4.1 P-44ESect-4.1 P-45ESect-4.1 P-46ESect-4.1 P-47ESect-4.1 P-48ESect-4.1 P-49ESect-4.1 P-50ESect-4.1 P-51ESect-4.1 P-52ESect-4.1 P-53ESect-4.1 P-54ESect-4.1 P-55ESect-4.1 P-56ESect-4.1 P-57ESect-4.1 P-58ESect-4.1 P-59ESect-4.1 P-60ESect-4.1 P-61ESect-4.1 P-62ESect-4.1 P-63ESect-4.1 P-64ESect-4.1 P-65ESect-4.1 P-66ESect-4.1 P-67ESect-4.1 P-68ESect-4.1 P-69ESect-4.1 P-70ESect-4.1 P-71ESect-4.1 P-72ESect-4.1 P-73ESect-4.1 P-74ESect-4.1 P-75ESect-4.1 P-76ESect-4.1 P-77ESect-4.1 P-78ESect-4.1 P-79ESect-4.1 P-80ESect-4.2 P-1ESect-4.2 P-2ESect-4.2 P-3ESect-4.2 P-4ESect-4.2 P-5ESect-4.2 P-6ESect-4.2 P-7ESect-4.2 P-8ESect-4.2 P-9ESect-4.2 P-10ESect-4.2 P-11ESect-4.2 P-12ESect-4.2 P-13ESect-4.2 P-14ESect-4.2 P-15ESect-4.2 P-16ESect-4.2 P-17ESect-4.2 P-18ESect-4.2 P-19ESect-4.2 P-20ESect-4.2 P-21ESect-4.2 P-22ESect-4.2 P-23ESect-4.2 P-24ESect-4.2 P-25ESect-4.2 P-26ESect-4.2 P-27ESect-4.2 P-28ESect-4.2 P-29ESect-4.2 P-30ESect-4.2 P-31ESect-4.2 P-32ESect-4.2 P-33ESect-4.2 P-34ESect-4.2 P-36ESect-4.2 P-37ESect-4.2 P-38ESect-4.3 P-1ESect-4.3 P-2ESect-4.3 P-3ESect-4.3 P-4ESect-4.3 P-5ESect-4.3 P-6ESect-4.3 P-7ESect-4.3 P-8ESect-4.3 P-9ESect-4.3 P-10ESect-4.3 P-11ESect-4.3 P-12ESect-4.3 P-13ESect-4.3 P-14ESect-4.3 P-15ESect-4.3 P-16ESect-4.3 P-17ESect-4.3 P-18ESect-4.3 P-19ESect-4.3 P-20ESect-4.3 P-21ESect-4.3 P-22ESect-4.3 P-23ESect-4.3 P-24ESect-4.3 P-25ESect-4.3 P-26ESect-4.3 P-27ESect-4.3 P-28ESect-4.3 P-29ESect-4.3 P-30ESect-4.3 P-31ESect-4.3 P-32ESect-4.3 P-33ESect-4.3 P-34ESect-4.3 P-35ESect-4.3 P-36ESect-4.3 P-37ESect-4.3 P-38ESect-4.3 P-39ESect-4.3 P-40ESect-4.3 P-41ESect-4.3 P-42ESect-4.3 P-43ESect-4.3 P-44ESect-4.3 P-45ESect-4.3 P-46ESect-4.3 P-47ESect-4.3 P-48ESect-4.3 P-49ESect-4.3 P-50ESect-4.3 P-51ESect-4.3 P-52ESect-4.3 P-53ESect-4.3 P-54ESect-4.3 P-55ESect-4.3 P-56ESect-4.3 P-57ESect-4.3 P-58ESect-4.3 P-59ESect-4.3 P-60ESect-4.3 P-61ESect-4.3 P-62ESect-4.3 P-66ESect-4.3 P-67ESect-4.3 P-68ESect-4.3 P-69ESect-4.3 P-70ESect-4.3 P-71ESect-4.3 P-72ESect-4.3 P-73ESect-4.3 P-74ESect-4.3 P-75ESect-4.3 P-76ESect-4.3 P-77ESect-4.3 P-78ESect-4.3 P-79ESect-4.3 P-80ESect-4.3 P-81ESect-4.3 P-82ESect-4.3 P-83ESect-4.3 P-84ESect-4.3 P-85ESect-4.3 P-86ESect-4.3 P-87ESect-4.3 P-88ESect-4.3 P-89ESect-4.3 P-90ESect-4.3 P-92ESect-4.3 P-93ESect-4.4 P-1ESect-4.4 P-2ESect-4.4 P-3ESect-4.4 P-4ESect-4.4 P-5ESect-4.4 P-6ESect-4.4 P-7ESect-4.4 P-8ESect-4.4 P-9ESect-4.4 P-10ESect-4.4 P-11ESect-4.4 P-12ESect-4.4 P-13ESect-4.4 P-14ESect-4.4 P-15ESect-4.4 P-16ESect-4.4 P-17ESect-4.4 P-18ESect-4.4 P-19ESect-4.4 P-20ESect-4.4 P-21ESect-4.4 P-22ESect-4.4 P-23ESect-4.4 P-24ESect-4.4 P-25ESect-4.4 P-26ESect-4.4 P-27ESect-4.4 P-28ESect-4.4 P-29ESect-4.4 P-30ESect-4.4 P-31ESect-4.4 P-32ESect-4.4 P-33ESect-4.4 P-34ESect-4.4 P-35ESect-4.4 P-36ESect-4.4 P-37ESect-4.4 P-38ESect-4.4 P-39ESect-4.4 P-40ESect-4.4 P-41ESect-4.4 P-42ESect-4.4 P-43ESect-4.4 P-44ESect-4.4 P-45ESect-4.4 P-46ESect-4.4 P-47ESect-4.4 P-48ESect-4.4 P-49ESect-4.4 P-50ESect-4.4 P-51ESect-4.4 P-52ESect-4.4 P-53ESect-4.4 P-54ESect-4.4 P-55ESect-4.4 P-56ESect-4.4 P-57ESect-4.4 P-58ESect-4.4 P-59ESect-4.4 P-60ESect-4.4 P-61ESect-4.4 P-62ESect-4.4 P-63ESect-4.4 P-64ESect-4.4 P-65ESect-4.4 P-66ESect-4.4 P-67ESect-4.4 P-68ESect-4.4 P-69ESect-4.4 P-70ESect-4.4 P-71ESect-4.4 P-72ESect-4.4 P-73ESect-4.4 P-74ESect-4.4 P-75ESect-4.4 P-76ESect-4.4 P-77ESect-4.4 P-78ESect-4.4 P-79ESect-4.4 P-80ESect-4.4 P-81ESect-4.4 P-82ESect-4.4 P-83ESect-4.4 P-84ESect-4.4 P-85ESect-4.4 P-86ESect-4.4 P-87ESect-4.4 P-88ESect-4.4 P-89ESect-4.4 P-90ESect-4.4 P-91ESect-4.4 P-92ESect-4.5 P-1ESect-4.5 P-2ESect-4.5 P-3ESect-4.5 P-4ESect-4.5 P-5ESect-4.5 P-6ESect-4.5 P-7ESect-4.5 P-8ESect-4.5 P-9ESect-4.5 P-10ESect-4.5 P-11ESect-4.5 P-12ESect-4.5 P-13ESect-4.5 P-14ESect-4.5 P-15ESect-4.5 P-16ESect-4.5 P-17ESect-4.5 P-18ESect-4.5 P-19ESect-4.5 P-20ESect-4.5 P-21ESect-4.5 P-22ESect-4.5 P-23ESect-4.5 P-24ESect-4.5 P-25ESect-4.5 P-26ESect-4.5 P-27ESect-4.5 P-28ESect-4.5 P-29ESect-4.5 P-30ESect-4.5 P-31ESect-4.5 P-32ESect-4.5 P-33ESect-4.5 P-34ESect-4.5 P-35ESect-4.5 P-36ESect-4.5 P-37ESect-4.5 P-38ESect-4.5 P-39ESect-4.5 P-40ESect-4.5 P-41ESect-4.5 P-42ESect-4.5 P-43ESect-4.5 P-44ESect-4.5 P-45ESect-4.5 P-46ESect-4.5 P-47ESect-4.5 P-48ESect-4.5 P-49ESect-4.5 P-50ESect-4.5 P-51ESect-4.5 P-52ESect-4.5 P-53ESect-4.5 P-54ESect-4.5 P-55ESect-4.5 P-56ESect-4.5 P-57ESect-4.5 P-58ESect-4.5 P-59ESect-4.5 P-60ESect-4.5 P-61ESect-4.5 P-62ESect-4.5 P-63ESect-4.5 P-64ESect-4.5 P-65ESect-4.5 P-66ESect-4.5 P-67ESect-4.5 P-68ESect-4.5 P-69ESect-4.5 P-70ESect-4.5 P-71ESect-4.5 P-72ESect-4.5 P-73ESect-4.5 P-74ESect-4.5 P-75ESect-4.5 P-76ESect-4.6 P-1ESect-4.6 P-2ESect-4.6 P-3ESect-4.6 P-4ESect-4.6 P-5ESect-4.6 P-6ESect-4.6 P-9ESect-4.6 P-10ESect-4.6 P-13ESect-4.6 P-14ESect-4.6 P-27ESect-4.6 P-28ESect-4.6 P-29ESect-4.6 P-30ESect-4.6 P-33ESect-4.6 P-34ESect-4.6 P-35ESect-4.6 P-38ESect-4.6 P-39ESect-4.6 P-40ESect-4.7 P-1ESect-4.7 P-2ESect-4.7 P-3ESect-4.7 P-4ESect-4.7 P-5ESect-4.7 P-6ESect-4.7 P-7ESect-4.7 P-8ESect-4.7 P-9ESect-4.7 P-10ESect-4.7 P-11ESect-4.7 P-12ESect-4.7 P-13ESect-4.7 P-14ESect-4.7 P-15ESect-4.7 P-16ESect-4.7 P-17ESect-4.7 P-18ESect-4.7 P-19ESect-4.7 P-20ESect-4.7 P-21ESect-4.7 P-22ESect-4.7 P-23ESect-4.7 P-24ESect-4.7 P-25ESect-4.7 P-26ESect-4.7 P-27ESect-4.7 P-28ESect-4.7 P-29ESect-4.7 P-30ESect-4.7 P-31ESect-4.7 P-32ESect-4.7 P-33ESect-4.7 P-34ESect-4.7 P-35ESect-4.7 P-36ESect-4.7 P-37ESect-4.7 P-38ESect-4.7 P-39ESect-4.7 P-40ESect-4.7 P-41ESect-4.7 P-42ESect-4.7 P-43ESect-4.7 P-44ESect-4.7 P-45ESect-4.7 P-46ESect-4.7 P-47ESect-4.7 P-48ESect-4.7 P-49ESect-4.7 P-50ESect-4.7 P-51ESect-4.7 P-52ESect-4.7 P-53ESect-4.7 P-54ESect-4.7 P-55ESect-4.7 P-56ESect-4.7 P-57ESect-4.7 P-58ESect-4.7 P-59ESect-4.7 P-60ESect-4.7 P-61ESect-4.7 P-62ESect-4.7 P-63ESect-4.7 P-64ESect-4.7 P-65ESect-4.7 P-66ESect-4.7 P-67ESect-4.7 P-68ESect-4.7 P-69ESect-4.7 P-70ESect-4.7 P-71ESect-4.7 P-72ESect-4.7 P-73ESect-4.7 P-74ESect-4.7 P-75ESect-4.7 P-76ESect-4.7 P-77ESect-4.7 P-78ESect-4.7 P-79ESect-4.7 P-80ESect-4.7 P-81ESect-4.7 P-82ESect-4.8 P-1ESect-4.8 P-2ESect-4.8 P-3ESect-4.8 P-4ESect-4.8 P-5ESect-4.8 P-6ESect-4.8 P-7ESect-4.8 P-8ESect-4.8 P-9ESect-4.8 P-10ESect-4.8 P-11ESect-4.8 P-12ESect-4.8 P-13ESect-4.8 P-14ESect-4.8 P-15ESect-4.8 P-16ESect-4.8 P-17ESect-4.8 P-18ESect-4.8 P-19ESect-4.8 P-20ESect-4.8 P-21ESect-4.8 P-22ESect-4.8 P-23ESect-4.8 P-24ESect-4.8 P-25ESect-4.8 P-26ESect-4.8 P-27ESect-4.8 P-29ESect-4.8 P-30ESect-4.8 P-32ESect-4.8 P-33ESect-4.8 P-34ESect-4.8 P-35ESect-4.8 P-36ESect-4.8 P-37ESect-4.8 P-39ESect-4.8 P-40ESect-4.8 P-41ESect-4.8 P-42ESect-4.9 P-1ESect-4.9 P-2ESect-4.9 P-3ESect-4.9 P-4ESect-4.9 P-5ESect-4.9 P-6ESect-4.9 P-7ESect-4.9 P-8ESect-4.9 P-9ESect-4.9 P-10ESect-4.9 P-11ESect-4.9 P-12ESect-4.9 P-13ESect-4.9 P-14ESect-4.9 P-15ESect-4.9 P-16ESect-4.9 P-17ESect-4.9 P-18ESect-4.9 P-19ESect-4.9 P-20ESect-4.9 P-21ESect-4.9 P-22ESect-4.9 P-25ESect-4.9 P-26ESect-4.9 P-27ESect-4.9 P-28ESect-4.9 P-29ESect-4.9 P-30ESect-4.9 P-31ESect-4.9 P-32ESect-4.9 P-33ESect-4.9 P-34ESect-4.9 P-35ESect-4.9 P-36ESect-4.9 P-37ESect-4.9 P-38ESect-4.9 P-39ESect-4.9 P-40ESect-4.9 P-41ESect-4.9 P-42ESect-4.9 P-43ESect-4.9 P-44ESect-4.9 P-45ESect-4.9 P-46ESect-4.9 P-47ESect-4.9 P-48ESect-4.9 P-49ESect-4.9 P-50ESect-4.9 P-51ESect-4.9 P-52ESect-4.9 P-53ESect-4.9 P-54ESect-4.9 P-55ESect-4.9 P-56ESect-4.9 P-57ESect-4.9 P-58ESect-4.9 P-59ESect-4.9 P-60ESect-4.9 P-61ESect-4.9 P-62ESect-4.9 P-63ESect-4.9 P-64ESect-4.9 P-65ESect-4.9 P-66ESect-4.9 P-67ESect-4.9 P-68ESect-4.9 P-69ESect-4.9 P-70ESect-4.9 P-71ESect-4.9 P-72ESect-4.9 P-73ESect-4.9 P-74ESect-4.9 P-75ESect-4.9 P-76ESect-4.9 P-77ESect-4.9 P-79ECh-4 P-1RCCCh-4 P-2RCCCh-4 P-3RCCCh-4 P-4RCCCh-4 P-5RCCCh-4 P-6RCCCh-4 P-7RCCCh-4 P-8RCCCh-4 P-9RCCCh-4 P-10RCCCh-4 P-11RCCCh-4 P-1RQCh-4 P-2RQCh-4 P-3RQCh-4 P-4RQCh-4 P-5RQCh-4 P-6RQCh-4 P-7RQCh-4 P-8RQCh-4 P-9RQCh-4 P-10RQCh-4 P-11RQCh-4 P-12RQCh-4 P-13RQCh-4 P-14RQCh-4 P-15RQCh-4 P-16RQCh-4 P-17RQCh-4 P-18RQCh-4 P-19RQCh-4 P-20RQCh-4 P-21RQCh-4 P-1RECh-4 P-2RECh-4 P-3RECh-4 P-4RECh-4 P-5RECh-4 P-6RECh-4 P-7RECh-4 P-8RECh-4 P-9RECh-4 P-10RECh-4 P-11RECh-4 P-12RECh-4 P-13RECh-4 P-14RECh-4 P-15RECh-4 P-16RECh-4 P-17RECh-4 P-18RECh-4 P-19RECh-4 P-20RECh-4 P-21RECh-4 P-22RECh-4 P-23RECh-4 P-24RECh-4 P-25RECh-4 P-26RECh-4 P-27RECh-4 P-28RECh-4 P-29RECh-4 P-30RECh-4 P-31RECh-4 P-32RECh-4 P-33RECh-4 P-34RECh-4 P-35RECh-4 P-36RECh-4 P-37RECh-4 P-38RECh-4 P-39RECh-4 P-43RECh-4 P-44RECh-4 P-45RECh-4 P-46RECh-4 P-47RECh-4 P-48RECh-4 P-49RECh-4 P-50RECh-4 P-51RECh-4 P-52RECh-4 P-53RECh-4 P-54RECh-4 P-55RECh-4 P-56RECh-4 P-57RECh-4 P-58RECh-4 P-59RECh-4 P-60RECh-4 P-61RECh-4 P-62RECh-4 P-63RECh-4 P-64RECh-4 P-65RECh-4 P-66RECh-4 P-67RECh-4 P-68RECh-4 P-69RECh-4 P-70RECh-4 P-71RECh-4 P-72RECh-4 P-73RECh-4 P-74RECh-4 P-75RECh-4 P-76RECh-4 P-77RECh-4 P-78RECh-4 P-79RECh-4 P-80RECh-4 P-81RECh-4 P-82RECh-4 P-83RECh-4 P-84RECh-4 P-85RECh-4 P-86RECh-4 P-1PCh-4 P-2PCh-4 P-3PCh-4 P-4PCh-4 P-5PCh-4 P-6PCh-4 P-7PCh-4 P-8PCh-4 P-9PCh-4 P-10PCh-4 P-11PCh-4 P-12PCh-4 P-13PCh-4 P-14PCh-4 P-15PCh-4 P-16PCh-4 P-17PCh-4 P-18PCh-4 P-19PCh-4 P-20PCh-4 P-21PCh-4 P-22PCh-4 P-23PCh-4 P-25PCh-4 P-26P

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Define the terms population and sample, and explain the role of each in a research study.

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Evaluate the integral. tan2xcos3xdx

Calculus (MindTap Course List)

In Exercises 21-24, find the distance between the given points. 21. (1, 3) and (4, 7)

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Determine whether the series converges or diverges. 16. n=11nn

Single Variable Calculus: Early Transcendentals

Distance Find the distance between the planes 5x3y+z=2 and 5x3j+z=3.

Calculus: Early Transcendental Functions (MindTap Course List)

True or False: x2x24dx should be evaluated using integration by parts.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The polar coordinates of the point P in the figure at the right are:

Study Guide for Stewart's Multivariable Calculus, 8th