Problem 34 - Algebra 2

Question

In a physics experiment, a steel ball on a flat track is accelerated and then allowed to roll freely. After the first minute, the ball has rolled 120 feet. Each minute the ball travels only 40% as far as it did during the preceding minute. How far does the ball travel?

Step-by-Step Solution

Answer

The ball travels a total of 200 feet.

Step 1: Understand the Problem

We are given that a steel ball travels a certain distance in a sequence of decreasing distances over minutes. In the first minute, the ball rolls 120 feet. In each subsequent minute, it rolls 40% of the distance it rolled the previous minute. We need to determine the total distance traveled as it continues to roll.

Step 2: List the Distances Rolled Each Minute

The sequence starts with 120 feet in the first minute. In the second minute, the ball travels 40% of 120 feet which can be calculated as \( 120 \times 0.4 \). The third minute is 40% of the second minute's distance: \( (120 \times 0.4) \times 0.4 \). The sequence continues this way for further minutes.

Step 3: Identify the Pattern

The distances traveled forms a geometric sequence where the first term \( a = 120 \) feet and the common ratio \( r = 0.4 \). Each subsequent term is given by \( a imes r^n \), where \( n \) is the number of the minute (or term index starting from 0 for the first term).

Step 4: Find the Sum of the Infinite Geometric Series

The total distance is the sum of this geometric series. For a geometric series with first term \( a \) and common ratio \( r \), the infinite sum \( S \) is given by \[ S = \frac{a}{1 - r} \text{ if } |r| < 1 \]. Substituting the values, we get \[ S = \frac{120}{1 - 0.4} = \frac{120}{0.6} = 200. \]

Step 5: Conclusion

The ball travels a total of 200 feet as it continues to roll indefinitely with each minute contributing a reducing distance.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero number. This fixed number is known as the "common ratio." In a geometric sequence, the rate of change between terms is consistent. This distinguishes geometric sequences from arithmetic ones, where each term is the previous term plus a fixed number.
For example, in the given physics experiment, the steel ball rolls a certain distance each minute. The distances form a geometric sequence. If the first distance rolled is 120 feet, and each subsequent distance is 40% of the last, then the distances the ball travels are 120 feet, followed by 48 feet (which is 120 x 0.4), and so on.

• The first term (\(a\)) is the first distance rolled, which is 120 feet.
• The pattern of multiplying by a constant (common ratio) defines a geometric sequence.

Geometric sequences are helpful in physics and engineering, where exponential growth, decay, or reduction patterns are involved.

Common Ratio

The common ratio is a key component of any geometric sequence. It is the factor by which we multiply each term to get the next term in the sequence. In mathematical terms, for a sequence with terms \(a_1, a_2, a_3, \ldots\), the common ratio \(r\) is given by \(r = \frac{a_2}{a_1}\).
For our physics problem, the common ratio \(r\) is 0.4. This means each subsequent distance that the steel ball travels is 40% of the previous distance. Understanding the common ratio helps predict future terms in the sequence.

• The value of the common ratio determines how quickly the terms in the sequence multiply or divide.
• If \(|r| < 1\), the sequence's terms decrease, leading to a convergence when summed infinitely.

Recognizing the common ratio allows us to determine behavior patterns in sequences, such as growth or decay in practical scenarios like bouncing balls or bank interests.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Calculating the sum of an infinite series in the context of geometric sequences provides insights into cumulative behaviors over time. If the common ratio \(|r|\) of a geometric series satisfies \(|r| < 1\), the series converges.
In our experiment, the steel ball rolling's total distance can be calculated using this concept. The series formed by the distances each minute converges because the common ratio is 0.4, which is less than 1. The formula for the sum \(S\) of an infinite series with a first term \(a\) and a common ratio \(r\) is: \[ S = \frac{a}{1 - r} \] Applying it to our sequence:

• The first term \(a\) is 120 feet.
• The common ratio \(r\) is 0.4.
• Thus, the sum of the series \(S = \frac{120}{1 - 0.4} = 200\).

This means the ball will eventually travel a total of 200 feet as it rolls indefinitely.

Physics Experiment

In a physics experiment involving motion, understanding the resulting patterns like geometric sequences is vital. For our hypothetical situation, a ball is allowed to roll freely after limited acceleration. Such experiments help study the effects of forces over time. By modeling the ball's rolling distance via geometric sequences, we can analyze the influence of friction and other decaying forces.
When conducting similar experiments, note:

• Initial conditions like initial velocity or starting distance impact the sequence.
• The rolling distances reveal how forces like friction slow the ball over time.
• Identifying the common ratio aids in predicting the ball's behavior and understanding physical principles at play.

Often in physics experiments, a mathematical model helps predict outcomes. Here, geometric sequences model the phenomenon, explaining eventual culmination of distance due to diminishing returns from friction.