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°.
- (a) Show that the radius r of the common face is given by
[Hint: Use the Law of Sines together with the fact that an angle θ and its supplement 180° − θ have the same sine.]
- (b) Find the radius of the common face if the radii of the bubbles are 4 cm and 3 cm.
- (c) What shape does the common face take if the two bubbles have equal radii?
(a)
To show: The radius of common face formed, when two bubbles cling together is given as,
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,
Figure (1)
It is given that
Let assume the unknown angle
Angles
That is
And angle
Substitute
That is,
Similarly angle
Substitute the value of
That is,
Draw the figure of triangle and show above angles.
Figure (1)
The law of sine for triangle,
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
Substitute
That is,
Similarly apply law of sine in
Add both the equations (5) and (6).
Use the trigonometric identity
Further simplify the equation as,
Hence, the radius of common face formed is,
(b)
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,
Substitute 4 for a and 3 for b.
Thus, the radius of common face is 12 cm.
(c)
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,
It is given that bubbles have same radii. Assume the radii of bubbles as
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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