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. Identify the type of variation: The problem states that the pitch varies inversely with the wavelength, which means we can use the formula $p = \frac{k}{w}$, where $p$ is the pitch, $w$ is the wavelength, and $k$ is the constant of variation.
2. Use the given values to find the constant of variation $k$: Substitute $p = 660$ and $w = 1.6$ into the formula to get $660 = \frac{k}{1.6}$. Solve for $k$ by multiplying both sides by 1.6.
3. Now that you have the constant $k$, use it to find the pitch of the tone with a wavelength of 2.4 feet: Substitute $w = 2.4$ into the formula $p = \frac{k}{w}$.
4. Simplify the expression to find the new pitch: Substitute the value of $k$ you found in step 2 into the equation from step 3.
5. Conclude by stating the relationship between pitch and wavelength: As the wavelength increases, the pitch decreases, which is consistent with the inverse variation relationship.

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 and light. 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 the context of pitch.

Proportional Relationships

Proportional relationships occur when two quantities maintain a constant ratio. In the context of the problem, the pitch of a tone and its wavelength are inversely proportional, meaning that as one increases, the other decreases. Recognizing this relationship allows for the application of proportional reasoning to find unknown values, such as determining the pitch for a different wavelength.