Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

Verified step by step guidance

1. Understand the problem: The pitch of a musical tone varies inversely as its wavelength. This means that as the wavelength increases, the pitch decreases, and vice versa. Mathematically, this relationship can be expressed as \( p = \frac{k}{w} \), where \( p \) is the pitch, \( w \) is the wavelength, and \( k \) is a constant of proportionality.
2. Use the given information to find the constant \( k \): Substitute the given pitch \( p = 660 \) vibrations per second and wavelength \( w = 1.6 \) feet into the formula \( p = \frac{k}{w} \). Solve for \( k \) by multiplying both sides of the equation by \( w \), resulting in \( k = p \cdot w \).
3. Substitute the values \( p = 660 \) and \( w = 1.6 \) into the equation \( k = p \cdot w \) to calculate the constant \( k \). This will give you the value of \( k \), which remains constant for this relationship.
4. Use the constant \( k \) to find the pitch of the tone with a wavelength of 2.4 feet. Substitute \( k \) and \( w = 2.4 \) into the formula \( p = \frac{k}{w} \). Solve for \( p \) by dividing \( k \) by \( w \).
5. Simplify the expression \( p = \frac{k}{w} \) to find the pitch of the tone with a wavelength of 2.4 feet. This will give you the final answer for the pitch.

Key Concepts

Inverse Variation

Inverse variation describes a relationship where one variable increases as the other decreases. In mathematical terms, if two variables x and y vary inversely, their product is constant (xy = k). This concept is crucial for solving problems where one quantity affects another in an opposite manner, such as the relationship between pitch and wavelength in this question.

Wavelength and Frequency

Wavelength and frequency are fundamental concepts in wave physics, particularly in sound. Wavelength is the distance between successive crests of a wave, while frequency refers to the number of vibrations or cycles per second, measured in hertz (Hz). Understanding how these two quantities relate is essential for solving problems involving sound waves, as they are inversely related in this context.

Setting Up Proportions

Setting up proportions is a method used to solve problems involving ratios and relationships between quantities. In this case, knowing the initial pitch and wavelength allows us to create a proportion to find the unknown pitch for a different wavelength. This technique is vital for applying the concept of inverse variation to find the solution effectively.