Problem 45 - Algebra 2

Question

An exhibit at a science museum offers visitors the opportunity to experiment with the motion of an object on a spring. One visitor pulled the object down and let it go. The object traveled a distance of 1.2 feet upward before heading back the other way. Each time the object changed direction, it moved only 80% as far as it did in the previous direction. Find the total distance the object traveled.

Step-by-Step Solution

Answer

The total distance traveled by the object is 6 feet.

Step 1: Understanding the problem

The object moves up and down in diminishing amounts until it stops. Each movement is 80% of the previous movement. This creates a geometric sequence where the first term is 1.2 feet.

Step 2: Identifying the geometric series

The sequence of distances traveled by the object is a geometric series with a first term (a) of 1.2 feet and a common ratio (r) of 0.8. The series is: 1.2, 0.96, 0.768, and so on.

Step 3: Calculation of the infinite series

The total distance traveled can be found using the formula for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \), where \(a = 1.2\) and \(r = 0.8\).

Step 4: Solve for total distance

Substitute \(a = 1.2\) and \(r = 0.8\) into the formula: \( S = \frac{1.2}{1 - 0.8} = \frac{1.2}{0.2} = 6 \) feet.

Key Concepts

Infinite Series

When dealing with an infinite series, you're looking at a sequence of numbers that continues indefinitely. Although it might seem counterintuitive, the sum of this series can actually amount to a finite number under certain conditions.
A great example of this is the geometric series, where each term in the series is a fixed fraction of the previous term. In the case of the object on the spring, it travels smaller and smaller distances each time, forming an infinite series.

• The first term in our example is 1.2 feet.
• Each subsequent term is 80% of the previous one, making this an infinite geometric series.

This decreasing pattern allows us to use a specific formula to find the sum of the entire series, despite its infinite nature: \[ S = \frac{a}{1 - r} \]This formula calculates a finite sum by considering the relationship between the first term and the common ratio.

Common Ratio

The common ratio is a key characteristic of a geometric sequence. It's the factor by which each term in the sequence is multiplied to produce the next term.
In the context of the spring motion problem, the common ratio is 0.8. This means each time the object changes direction, it travels 80% of the previous distance.

• The formula for the nth term in a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \) where \(a_1\) is the first term and \(r\) is the common ratio.
• In our example: the first term is 1.2, so the second term would be 1.2 × 0.8, the third term would be 0.96 × 0.8, and so forth.

Understanding the common ratio is essential because it determines the rate at which the terms of the sequence decrease.

Geometric Sequence

A geometric sequence is a series of numbers where each term is obtained by multiplying the previous one by a fixed, non-zero number known as the common ratio.
This type of sequence is recognized by the consistency in how each term is formed, which is why it can be so predictable and easy to work with when you know the starting term and the common ratio.

• In a geometric sequence, the relationship between consecutive terms is multiplicative rather than additive.
• The sequence in the spring problem starts with 1.2, followed by 0.96, then 0.768, and continues infinitely.

This sequential process highlights how geometric sequences can effectively model real-world scenarios, such as the diminishing motion of the object on the spring. By calculating the terms and using them in a geometric series, we can find solutions to problems involving seemingly never-ending sequences.