Problem 124 - Elementary and Intermediate Algebra

Question

The frequency of vibration of a string varies directly as the square root of the tension and inversely as the length of the string. Suppose a string 2.5 feet long, under a tension of 16 pounds, vibrates 25 times per second. Find \(k\) the constant of proportionality.

Step-by-Step Solution

Answer

The constant of proportionality \( k \) is 15.625.

Step 1: Understand the Relationship

The problem states that the frequency of vibration \( f \) varies directly as the square root of the tension \( T \) and inversely as the length \( L \). This can be expressed as the equation \( f = k \sqrt{T}/L \), where \( k \) is the constant of proportionality we need to find.

Step 2: Substitute Known Values

We are given that the frequency \( f = 25 \) vibrations per second, the tension \( T = 16 \) pounds, and the length \( L = 2.5 \) feet. Substitute these values into the equation: \( 25 = k \frac{\sqrt{16}}{2.5} \).

Step 3: Simplify the Expression

Calculate \( \sqrt{16} \), which equals 4. Then substitute back into the equation: \( 25 = k \frac{4}{2.5} \).

Step 4: Solve for \( k \)

Simplify \( \frac{4}{2.5} \) to get \( 1.6 \). The equation becomes \( 25 = k \times 1.6 \). To isolate \( k \), divide both sides by 1.6: \( k = \frac{25}{1.6} \).

Step 5: Compute the Value of \( k \)

Perform the division: \( \frac{25}{1.6} = 15.625 \). Hence, \( k = 15.625 \).

Key Concepts

Direct Variation Unveiled

In mathematics, direct variation describes a simple relationship between two variables. When a variable changes, the other one changes at a constant rate. For example, if the frequency of a string’s vibration varies directly with the square root of the tension, it means:

• If you increase the tension in the string, the frequency of vibration goes up.
• If you decrease the tension, the frequency goes down.

The key idea here is proportionality. The change in frequency is proportional to the change in the square root of tension. Hence, the direct variation can be expressed as:
\[f = k \sqrt{T} \quad \text{(where \( k \) is the constant of proportionality)}\]This equation tells you that as the tension ( \( T \)) becomes larger, so does \( f \), as long as the constant \( k \) remains the same. This simplistic relationship has real-world applications, making it crucial for students to master the concept.

Exploring Inverse Variation

Inverse variation is when one variable increases while the other decreases, maintaining a constant relationship. A classic example is seen in the frequency of a string, which varies inversely with its length. This can be encapsulated by the formula:
\[f = \frac{k}{L} \quad \text{(where \( k \) is the constant of proportionality)}\]This means:

• If the length of the string increases, the frequency of vibration decreases.
• If the length decreases, then the frequency increases.

Think of this like a seesaw in physics.
In the given problem, if you stretch the string by making it longer, the string vibrates slower, assuming tension is constant. This balancing act between two variables underpins inverse variation. Recognizing this pattern helps in solving complex math problems and understanding everyday phenomena.

Mathematical Relationships with Proportionality

Understanding mathematical relationships is key to deciphering problems like the one given. Direct and inverse variations are both rooted in proportionality, where one value affects another in predictable ways.
When analyzing such problems, the equation for frequency:
\[f = k \frac{\sqrt{T}}{L}\]illustrates how two forms of variation combine.

• Direct in numerator \( (\sqrt{T}) \)
• Inverse in denominator \( (L) \)

This multifaceted relationship shows how complex interactions lead to simple outcomes like the frequency of vibration.
Solving such problems often means:

• Understanding the fundamental theory
• Substituting known values
• Isolating and solving for the unknown (\( k \))

Grasping these concepts enriches your problem-solving toolkit, allowing you to view not just math but real life through a lens of logical and proportional interactions. Whether in science, engineering, or daily life, these principles illuminate the interconnected world around you.