Problem 87 - Algebra and Trigonometry

Question

A pendulum swings through an arc of length 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 10 th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length has the pendulum swung? (d) When it stops, what total length has the pendulum swung?

Step-by-Step Solution

Answer

a) 0.7748 b) 8th swing c) 14.9096 feet d) 20 feet

Step 1: Identify the initial term and common ratio

The initial length of the arc is 2 feet, and each swing is 0.9 of the previous length, making the common ratio 0.9.

Step 2: Find the length of the arc of the 10th swing

Use the formula for the nth term of a geometric sequence: \( a_n = a_1 \times r^{n-1} \)For the 10th term, \( a_{10} = 2 \times 0.9^{9} \). Calculate \( 0.9^9 \) and then multiply by 2.

Step 3: Determine on which swing the arc length is first less than 1 foot

For arc length less than 1 foot, solve the inequality: \( 2 \times 0.9^{n-1} < 1 \), or \( 0.9^{n-1} < 0.5 \). Take the natural logarithm of both sides: \( \text{ln}(0.9^{n-1}) < \text{ln}(0.5) \), which simplifies to \( (n-1) \text{ln}(0.9) < \text{ln}(0.5) \). Solve for \( n \).

Step 4: Calculate the total length swung after 15 swings

Use the formula for the sum of the first n terms of a geometric series: \( S_n = a_1 \frac{1-r^n}{1-r} \). Here, \( S_{15} = 2 \frac{1-0.9^{15}}{1-0.9} \). Calculate \( 0.9^{15} \) and then substitute back into the equation.

Step 5: Determine the total length when the pendulum stops

Find the sum to infinity of the geometric series using the formula: \( S = \frac{a_1}{1-r} \). Here, \( S = \frac{2}{1-0.9} \).

Key Concepts

Initial Term

Understanding the initial term in a geometric sequence is key. The initial term, commonly denoted as \( a_1 \), is the first number in the sequence. In the context of our pendulum problem:

• The initial term \( a_1 \) is the initial swing length, which is 2 feet.

Always remember: the initial term sets the baseline for all following terms.

Grasp this concept well because subsequent terms depend on it!

Common Ratio

The common ratio in a geometric sequence (denoted as \( r \)) is the factor by which each term is multiplied to get the next term.

• For our pendulum example, the common ratio is 0.9 since each swing is 0.9 of the previous one.

Think of the common ratio as the multiplier that scales down or up the terms in your sequence.
It’s essential for calculating further terms and sums in geometric sequences.

Geometric Series Sum Formula

To find the sum of a geometric series up to a certain term or when it continues infinitely, we use specific formulas:

• Finite Sum Formula (Sum of the first n terms): \[ S_n = a_1 \frac{1-r^n}{1-r} \text{\ For our pendulum after 15 swings: } S_{15} = 2 \frac{1-0.9^{15}}{1-0.9} \]

• Infinite Sum Formula: When \|r\| < 1, the series sum converges: \[ S = \frac{a_1}{1-r} \text{\ For our pendulum: } S = \frac{2}{1-0.9} \]

These formulas are crucial for solving problems involving the total distance swung by the pendulum.

Inequality Solving

Inequality solving helps find out where a certain term in a sequence meets specific conditions:

• To find when the pendulum's arc length is less than 1 foot, we solve: \( 2 \times 0.9^{n-1} < 1 \)

Steps:

• Apply logs to linearize: \( \text{ln}(0.9^{n-1}) < \text{ln}(0.5) \)
• Simplify to get: \( (n-1) \text{ln}(0.9) < \text{ln}(0.5) \)
• Solve for \( n \): \( n-1 > \frac{\text{ln}(0.5)}{\text{ln}(0.9)} \rightarrow n = ... \)

This method helps pinpoint the exact term meeting our condition.