Problem 53 - College Algebra

Question

For the following exercises, use the given information to answer the questions. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

Step-by-Step Solution

Answer

The string should be 48 inches long.

Step 1: Understand Inverse Variation

The problem states that the rate of vibration varies inversely with the length of the string. This can be represented as \( V = \frac{k}{L} \), where \( V \) is the rate of vibration (vibrations per second), \( L \) is the length of the string, and \( k \) is a constant.

Step 2: Find the Constant k

Given that a string 24 inches long vibrates at 128 times per second, we substitute these values into the formula: \( 128 = \frac{k}{24} \). Solving for \( k \), we get \( k = 128 \times 24 = 3072 \).

Step 3: Set Up the Equation for the Unknown Length

We are asked to find the length \( L \) of a string that vibrates 64 times per second. Using the inverse variation formula again: \( 64 = \frac{3072}{L} \).

Step 4: Solve for the New Length

To find \( L \), rearrange the equation: \( L = \frac{3072}{64} \). Simplifying, we get \( L = 48 \). Thus, the length of the string is 48 inches.

Key Concepts

Rate of Vibration

The rate of vibration refers to how frequently a string vibrates over a period, typically measured in vibrations per second. This concept is crucial because it relates to the pitch or frequency of the sound produced by the string. In musical terms, higher vibrations per second mean a higher pitch, while lower vibrations mean a deeper tone.

In our problem, we saw that the vibration rate of a string under constant tension was 128 times per second when the string was 24 inches long. When we decrease the rate of vibration to 64 times per second, we observe a relationship to the string's length. This illustrates how the rate of vibration is deeply interconnected with other factors such as the length of the string.

String Tension

String tension refers to how tightly a string is stretched. In real-world scenarios, tension significantly affects the sound produced by a string when it vibrates, as well as its rate of vibration. However, it's essential to note that in this specific exercise, tension is considered constant.

Holding tension constant allows us to focus solely on the inverse relationship between the string length and its rate of vibration. By understanding this isolated relationship, we can more easily derive formulas and predictions about how a string will behave when its length is changed, all other factors being stable.

Length of String

The length of the string is a crucial factor in determining the rate of vibration. According to the principle of inverse variation, as the string's length increases, the rate of vibration decreases and vice versa.

Here, we've seen that a string 24 inches long vibrates at a rate of 128 times per second. When the rate of vibration drops to 64 times per second, the length needs to be doubled to 48 inches to maintain that lower rate. This clearly demonstrates the inverse relationship: halving the rate of vibration results in doubling the string length when tension is held constant.

Constant of Variation

The constant of variation, denoted as \( k \) in inverse variation problems, plays a crucial role in establishing a relationship between two variables. It represents a fixed value that remains unchanged as the dependent and independent variables fluctuate.

In our exercise, \( k \) was computed as 3072 using the initial conditions: a 24-inch length vibrating 128 times per second. Once \( k \) is known, it simplifies predicting how changes in one variable will affect another, using the inverse variation formula \( V = \frac{k}{L} \).

This constant provides a stable foundation to calculate unknown values, enabling us to find the length a string must be to vibrate at a different rate without changing the tension.