Problem 55 - Precalculus with Calculus Previews

Question

The threshold of pain is generally taken to be around \(140 \mathrm{~dB}\). Find the intensity of sound \(I\) corresponding to \(140 \mathrm{~dB}\).

Step-by-Step Solution

Answer

The intensity is 100 W/m².

Step 1: Understand the Decibel Scale

The decibel (dB) scale is a logarithmic scale used to measure the intensity of sound. It is given by the formula: \( eta = 10 imes ext{log}_{10} rac{I}{I_0} \), where \( \beta \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 = 10^{-12} \mathrm{~W/m^2} \) is the reference intensity.

Step 2: Set Up the Equation for 140 dB

We know that the sound level \( \beta \) is \( 140 \mathrm{~dB} \). Substitute \( \beta = 140 \) into the equation: \( 140 = 10 \times ext{log}_{10} \frac{I}{10^{-12}} \).

Step 3: Simplify the Equation

Divide both sides by 10 to simplify: \( 14 = ext{log}_{10} \frac{I}{10^{-12}} \).

Step 4: Solve for Sound Intensity

To find \( I \), rewrite the equation in terms of \( I \): \( \frac{I}{10^{-12}} = 10^{14} \). Multiply both sides by \( 10^{-12} \): \( I = 10^{14} \times 10^{-12} \).

Step 5: Calculate the Intensity

Simplify the expression: \( I = 10^{2} = 100 \mathrm{~W/m^2} \). This is the intensity corresponding to \( 140 \mathrm{~dB} \).

Key Concepts

Decibel Scale

The decibel (dB) scale is a handy tool for expressing sound intensity levels. It's a logarithmic measure used primarily to compare the power levels of sound. Decibels are commonly used because they offer a more manageable range, translating large variations in sound intensity into smaller, more understandable numbers.
This scale is defined by the formula:

• \( \beta = 10 \times \log_{10} \frac{I}{I_0} \)

Where \( \beta \) represents the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity. Sound measured in decibels can communicate the intensity in a physical environment very effectively, as it directly correlates to the human perception of loudness.
Using the decibel scale for calculations involves logarithmic values, making it ideal for measuring extremely high or low sound intensities without using cumbersome figures.

Logarithmic Scale

A logarithmic scale is particularly useful for plotting data that covers a large range of values. Here, it helps convert the wide range of possible sound intensities into a more comprehensible scale. This scale makes it easier to compare the intensity of sounds that are vastly different in magnitude.
On the decibel scale:

• The relationship is logarithmic, meaning it corresponds to powers of ten.
• A 10 dB increase is perceived as approximately doubling of the loudness.

This inherently non-linear relation compresses large data ranges into smaller scales. As part of sound measurement, the logarithmic scale helps us understand phenomena that span many orders of magnitude by presenting them in manageable increments.
In context, the logarithmic scale transforms the threshold of pain (which is often at 140 dB) into a simple calculation involving exponents. This makes it far easier to conceptualize and work with than if we were dealing with the raw sound intensities directly.

Reference Intensity

Reference intensity, noted as \( I_0 \), serves as a baseline for sound comparisons. It is critical in calculating decibel levels and is set at \( 10^{-12} \text{W/m}^2 \), representing the faintest sound that can be heard by a typical human ear. This value is fundamental as a lower limit in sound intensity measurements for decibel calculations.

• The reference intensity allows us to quantify how much more intense a sound is than this baseline.
• Sound intensity values are expressed relative to this low threshold, allowing for normalized comparisons.

Understanding the reference intensity is key to working with the decibel scale. It grounds the decibel measurement in a physical reality, allowing each sound intensity to be precisely compared against a known baseline. The careful selection of this reference point facilitates consistent, reproducible sound level assessments that align with human hearing capabilities, thus ensuring meaningful outcomes when measuring sound loudness.