Concept explainers
To find:
a. The length of the arc of the 10th swing of the pendulum.
Answer to Problem 87AYU
a. The length of the arc of the 10th swing of the pendulum feet.
Explanation of Solution
Given:
It is given that the pendulum swings through an arc of 2 feet and on each successive swing the length of the arc is times the previous swing length.
The length of the first swing is 2 feet.
The length of the second swing is feet.
The length of the third swing is feet.
The length of swing sequence will be
Therefore, the length of the swing follows the geometric sequence.
a. The length of the arc of the 10th swing of the pendulum.
Let’s find the .
To find:
b. The swing number when the length of the arc first less than 1 foot.
Answer to Problem 87AYU
b. At 8th swing, the length of the arc is less than 1 foot
Explanation of Solution
Given:
It is given that the pendulum swings through an arc of 2 feet and on each successive swing the length of the arc is times the previous swing length.
The length of the first swing is 2 feet.
The length of the second swing is feet.
The length of the third swing is feet.
The length of swing sequence will be
Therefore, the length of the swing follows the geometric sequence.
b. The length of the arc of the nth swing is . For this to be exactly 1 foot requires that .
Divide both sides by 2.
Taking logarithm on both sides, .
At 8th swing, the length of the arc is less than 1 foot.
To find:
c. The total length of swing after 15 swings.
Answer to Problem 87AYU
c. The total length of swing after 15 swings feet
Explanation of Solution
Given:
It is given that the pendulum swings through an arc of 2 feet and on each successive swing the length of the arc is times the previous swing length.
The length of the first swing is 2 feet.
The length of the second swing is feet.
The length of the third swing is feet.
The length of swing sequence will be
Therefore, the length of the swing follows the geometric sequence.
c. The total length of swing after 15 swings.
After 15 swing, the pendulum will have swing the following total distance .
feet.
To find:
d. The total length travelled by the pendulum swing when it stops.
Answer to Problem 87AYU
d. The total length travelled by the pendulum swing when it stops feet.
Explanation of Solution
Given:
It is given that the pendulum swings through an arc of 2 feet and on each successive swing the length of the arc is times the previous swing length.
The length of the first swing is 2 feet.
The length of the second swing is feet.
The length of the third swing is feet.
The length of swing sequence will be
Therefore, the length of the swing follows the geometric sequence.
d. The total length travelled by the pendulum swing when it stops.
Pendulum swings stops means it converges to a particular point.
Sum of the infinite geometric series formula has to be used to find the total length travelled by the pendulum.
Convergence of an infinite geometric series theorem states that If converges. Its sum is .
Required total length feet.
Chapter 12 Solutions
Precalculus
Additional Math Textbook Solutions
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