Problem 67 - College Algebra

Question

A pendulum swings through an arc of 20 inches. On each successive swing, the length of the arc is \(90 \%\) of the previous length. After 10 swings, what is the total length of the distance the pendulum has swung?

Step-by-Step Solution

Answer

The total distance the pendulum has swung after 10 swings will be calculated by substituting into the formula for the sum of the terms of a geometric progression, obtaining the result using a decimal approximation.

Step 1: Convert Percentage to Decimal

90% is equivalent to \(0.9\). This will now be the common ratio of our geometric sequence having the successive length of swings.

Step 2: Determine the Form of the Sequence

The lengths of the swings create a geometric sequence where the first term (\(a\)) is 20 and the common ratio (\(r\)) is 0.9. So, the nth term in this sequence is given as \(a*r^{(n-1)}\).

Step 3: Find the Total Distance Swung After 10 Swings

To find the total distance, we need to sum the lengths of the 10 swings. The sum \(\(S_n\)\) of the first n terms in a geometric progression can be calculated with the formula \(S_n = a * (1 - r^n) / (1 - r)\), where n is the number of terms. Substituting our values into the formula, we find \(S_10 = 20 * (1 - 0.9^{10}) / (1 - 0.9)\).

Key Concepts

Pendulum Motion

Pendulum motion is an example of periodic motion that occurs when an object swings back and forth from a fixed point. Imagine a pendulum in a clock or one used for scientific demonstrations. It has regular, predictable movements due to gravity. The distance it covers while swinging can be fascinating to observe and calculate.

• The arc formed by the pendulum's swing is similar to a curve.
• Each swing might reduce due to friction, air resistance, or other forces.

In our exercise, the pendulum starts with a significant arc length, but this length decreases with each swing. Understanding how such motion can be described through sequences helps us see the regularity hidden in what might seem like chaotic motion.

Common Ratio

The concept of a common ratio is a fundamental part of geometric sequences. It denotes how each term in the sequence relates to the previous one. In our pendulum exercise, the common ratio is particularly revealing.

• Here, the common ratio is 0.9, meaning each subsequent swing is 90% the length of the previous one.
• This ratio shows a consistent pattern of reduction and helps mathematically model the sequence of movements.

Understanding the common ratio allows us to predict future values in the sequence without directly measuring them each time.

Sum of Terms

The sum of terms in a geometric sequence gives us the total accumulated motion or quantity over several iterations. For the pendulum problem, calculating the sum tells us the total distance the pendulum has swung after multiple swings.

• We sum the lengths of the arcs for a specified number of terms, here 10 swings.
• This calculation incorporates the decreasing lengths due to the common ratio, providing a cumulative total distance.

It's a powerful way to summarize what's happening over a series of events, combining both the magnitude of each swing and the total effort involved.

Geometric Progression Formula

A geometric progression formula provides a straightforward method to calculate the sum of terms in a sequence where each term is a fixed multiple of the previous one. In the context of our pendulum problem, this formula becomes quite useful.

• The formula for the sum of a geometric sequence is \[S_n = a * \frac{1 - r^n}{1 - r}\]
• Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
• It simplifies the problem of adding up all the terms manually, especially when there are many swings as in our example.

This efficient formula provides clear insight into how much total movement the pendulum achieves over time.