Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Textbook Question
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Chapter 6.5, Problem 44E

Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere whose center D lies on the line passing through the centers of the bubbles (see the figure). Also, ∠ACB and ∠ACD each have measure 60°.

  1. (a) Show that the radius r of the common face is given by

r = a b a b

[Hint: Use the Law of Sines together with the fact that an angle θ and its supplement 180° − θ have the same sine.]

  1. (b) Find the radius of the common face if the radii of the bubbles are 4 cm and 3 cm.
  2. (c) What shape does the common face take if the two bubbles have equal radii?

Chapter 6.5, Problem 44E, Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere

(a)

Expert Solution
Check Mark
To determine

To show: The radius of common face formed, when two bubbles cling together is given as, r=abab .

Explanation of Solution

To prove the radius of common face between two bubbles, assume one unknown angle as a variable and evaluate the angles with respect to variable.

The given figure is,

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 6.5, Problem 44E , additional homework tip  1

Figure (1)

It is given that ACB=60° and ACD=60° .

Let assume the unknown angle CAD be θ .

Angles CAD and CAB are in straight line. Thus,

CAD+CAB=180°θ+CAB=180°CAB=180°θ

That is CAB=180°θ . (1)

And angle CAD is an exterior angle of ΔCAB .

CAD=CAB+ACB

Substitute θ for CAD and 60° for ACB .

θ=CBA+60°CBA=θ60°

That is, CBA=θ60° . (2)

Similarly angle CAB is an exterior angle of ΔCAD .

CAB=ACD+CDA

Substitute the value of CAB from equation (1) and 60° for ACD .

180°θ=60°+CDACDA=180°θ60°CDA=120°θ

That is, CDA=120°θ . (3)

Draw the figure of triangle and show above angles.

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 6.5, Problem 44E , additional homework tip  2

Figure (1)

The law of sine for triangle,

sinXx=sinYy=sinZz (4)

Where,

  • X, Y and Z are the angles of triangle
  • x, y and z are the sides with respective to angles.

Apply law of sine in ΔABC . Substitute CAB for X, CBA for Y, ACB for Z, a for x, b for y and AB for z in equation (4).

sin(CAB)a=sin(CBA)b=sin(ACB)AB

Substitute 180°θ for CAB and θ60° CBA and 60° for ACB in the above formula.

sin(180°θ)a=sin(θ60°)b=sin(60°)ABsin(180θ)°a=sin(θ60)°b

That is, sin(180θ)°a=sin(θ60)°b . (5)

Similarly apply law of sine in ΔABC and gets an equation as,

sin(θ)°r=sin(120θ)°b (6)

Add both the equations (5) and (6).

sin(180θ)°a+sin(θ)°r=sin(θ60)°b+sin(120θ)°bsin(θ)°a+sin(θ)°r=sin(θ60)°+sin(120θ)°b[sin(180x)°=sin(x)°]sinθ°(1a+1r)=sin(θ60)°+sin(120θ)°b

Use the trigonometric identity Sinx+Siny=2sin(x+y2)cos(xy2) to simplify the equation further.

sinθ°(1a+1r)=2sin(θ60+120θ2)°cos(θ60120+θ2)°bsinθ°(1a+1r)=2sin(30)°cos(θ90)°bsinθ°(1a+1r)=2(12)cos((90θ))°b[sin30°=12]sinθ°(1a+1r)=cos(90θ)°b[cos(x)°=cosx°]

Further simplify the equation as,

sinθ°(1a+1r)=sinθ°b[cos(90θ)°=sinx°]1a+1r=1b1r=1b1ar=abba

Hence, the radius of common face formed is, r=abab .

(b)

Expert Solution
Check Mark
To determine

To find: The radius of common face when two bubbles cling together.

Answer to Problem 44E

The radius of common face is 12 cm.

Explanation of Solution

Given:

The radii of bubbles that cling together to form the common face are given as 4 and 3 cm respectively.

Calculation:

From part (a), the radius of common face formed, when two bubbles cling together is given as,

r=abab

Substitute 4 for a and 3 for b.

r=4×343=121=12cm

Thus, the radius of common face is 12 cm.

(c)

Expert Solution
Check Mark
To determine

The shape of common face formed by two bubbles of same radii.

Explanation of Solution

From part (a), the radius of common face formed, when two bubbles cling together is given as,

r=abab

It is given that bubbles have same radii. Assume the radii of bubbles as k . Substitute k for a and k for b in the above formula.

r=k×kkk=k20[undefined value]

The radius is not defined.

Thus, the shape of common face is a straight line.

Chapter 6 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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