a.
To find: A recursive model of geometric decay.
Given:
A pendulum initially swings through an arc of length 20 inches.
The length of the arc is 0.85 of the previous swing.
Calculation:
Here, the sequence of arc lengths of pendulum swings represents a geometric sequence with first term,
Now, the recursive model is
b.
To rewrite: The model in part (a) using an explicit formula.
Given:
Calculation:
Here, the sequence of arc lengths of pendulum swings represents a geometric sequence with first term,
Now, the explicit formula is
c.
To find: The approximate total distance the pendulum has swung after 11 swings.
111 inches.
Given:
A pendulum initially swings through an arc of length 20 inches.
The length of the arc is 0.85 of the previous swing.
Formula used:
The sum to
Calculation:
Here, the sequence of arc lengths of pendulum swings represents a geometric sequence with first term,
Now, the total distance in 11 swings is
Hence, the pendulum has swung a distance of 111 inches in 11 swings.
d.
To find: The approximate total distance the pendulum has swung when it stops.
133.3 inches.
Given:
A pendulum initially swings through an arc of length 20 inches.
The length of the arc is 0.85 of the previous swing.
Formula used:
The sum to infinite terms of a geometric sequence with first term
Calculation:
Here, the sequence of arc lengths of pendulum swings represents a geometric sequence with first term,
Now, the total distance is
Hence, the pendulum has swung a distance of 133.3 inches when it stops.
Chapter 14 Solutions
High School Math 2015 Common Core Algebra 2 Student Edition Grades 10/11
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