Problem 77 - Precalculus

Question

Matt likes to drive around campus in his classic Mustang with the stereo blaring. If his boom stereo has a sound intensity of \(120 \mathrm{dB}\), how many watts per square meter does the stereo emit?

Step-by-Step Solution

Answer

The stereo emits sound at an intensity of 1 watt per square meter.

Step 1: Understanding Sound Intensity and Decibels

The intensity level of sound measured in decibels (dB) is given by the formula \( L = 10 \, \log_{10}\left(\frac{I}{I_0}\right) \), where \( L \) is the sound level in decibels, \( I \) is the sound intensity in watts per square meter, and \( I_0 \) is the reference intensity, typically \( 10^{-12} \, \text{W/m}^2 \). In this problem, we are given \( L = 120 \, \text{dB} \).

Step 2: Rearranging the Sound Intensity Formula

First, rearrange the formula to solve for \( I \): \( \log_{10}\left(\frac{I}{I_0}\right) = \frac{L}{10} \). Hence, \( \frac{I}{I_0} = 10^{\frac{L}{10}} \).

Step 3: Substituting Known Values

Substitute \( L = 120 \, \text{dB} \) and \( I_0 = 10^{-12} \, \text{W/m}^2 \) into the equation: \( \frac{I}{10^{-12}} = 10^{12} \). This implies \( I = 10^{12} \times 10^{-12} \).

Step 4: Calculating the Result

Calculate \( I \) using the above expression: \( I = 10^{12} \times 10^{-12} = 1 \, \text{W/m}^2 \). This means the stereo emits sound at an intensity of 1 watt per square meter.

Key Concepts

Decibels

Understanding decibels is key to grasping sound intensity levels. Decibels, abbreviated as dB, are a unit of measurement for sound intensity. They provide a convenient way to represent very large or very small numbers. It's important because sound levels can vary dramatically, from the faint sound of rustling leaves to a roaring jet engine.

• Decibels measure the intensity of sound by comparing it to a reference level. This reference level is a standard to which all sound is compared.
• The human ear perceives sound intensity logarithmically, making decibels a suitable unit. That's why sound measured in decibels uses a logarithmic scale.
• The decibel level determines how loud a sound is perceived.

For instance, a sound intensity level of 120 dB, as in the exercise, is very loud, comparable to a rock concert or a threshold of pain for most people. Understanding decibels helps us quantify and understand sound intensity better.

Logarithmic Scale

The logarithmic scale is crucial for understanding decibels and sound intensity. In a logarithmic scale, instead of increasing linearly, values increase exponentially. This is particularly useful for representing values that differ over a large range.

• A 10 dB increase represents a tenfold increase in sound intensity.
• This scaling helps to manage and interpret extremely high and low values efficiently.
• A logarithmic scale mirrors how the human ear perceives changes in sound intensity.

This logarithmic nature is why a sound of 120 dB is much more intense than the sound level might initially suggest. By using a formula, we can express the intensity level, making 120 dB on the scale equivalent to a sound intensity many times greater than a lower dB level. This efficiently and effectively conveys the magnitude of sound forces.

Watts per Square Meter

Watts per square meter (W/m²) is a unit for measuring sound intensity. It describes how much sound power passes through a specific area. This concept is fundamental in acoustics and understanding sound measurements.

• Sound intensity is calculated as power per unit area.
• Watts measure power, while the area is measured in square meters.
• This unit allows for a precise expression of how powerful a sound is in a given space.

For example, an intensity level of 1 W/m², as found in the exercise, indicates a relatively high sound power, characteristic of very loud environments. This unit helps us assess environments' potential impacts on hearing and sets standards for safe sound exposure levels. Understanding watts per square meter demystifies how power translates into sound intensity.