Initially, a pendulum swings through an arc of 2 feet. On...

Name: Problem 87, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-03-20T16:04:55-08:00
Duration: 2 min 56 s
Channel: Sheryl Ezze
Description: Initially, a pendulum swings through an arc of 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length. (a) What is the length of the arc of the 10th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length will the pendulum have swung? (d) When it stops, what total length will the pendulum have swung?

Initially, a pendulum swings through an arc of 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length. (a) What is the length of the arc of the 10th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length will the pendulum have swung? (d) When it stops, what total length will the pendulum have swung?

Initially, a pendulum swings through an arc of 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length.
(a) What is the length of the arc of the 10th swing?
(b) On which swing is the length of the arc first less than 1 foot?
(c) After 15 swings, what total length will the pendulum have swung?
(d) When it stops, what total length will the pendulum have swung?

Key Concepts

Infinite Series Convergence

Infinite series convergence refers to the idea that when the common ratio of a geometric sequence is between -1 and 1, the infinite sum of the series converges to a finite number. This is an important concept when determining the total length traveled by the pendulum as it swings indefinitely, as the cumulative distance approaches a finite limit.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This concept is used to represent problems where quantities, such as the arc lengths of a pendulum, decrease (or increase) by a fixed percentage or factor with each step or swing.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. This concept is particularly useful for determining total quantities over a number of steps, such as calculating the total distance traveled by a pendulum over multiple swings or even over an infinite number of swings when the series converges.

Exponential Decay

Exponential decay describes a process where the quantity decreases at a rate proportional to its current value, typically characterized by a constant ratio between successive terms. This concept explains how the arc length of a pendulum steadily decreases by the same proportion with each swing.

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