Problem 42 - Precalculus Mathematics for Calculus

Question

These exercises deal with logarithmic scales. The noise from a power mower was measured at 106 dB. The noise level at a rock concert was measured at 120 dB. Find the ratio of the intensity of the rock music to that of the power mower.

Step-by-Step Solution

Answer

The intensity ratio of the rock music to the power mower is approximately 25.12.

Step 1: Understand the Decibel Scale

The decibel (dB) scale is a logarithmic scale used to measure the intensity of sound. Each increase of 10 dB represents a tenfold increase in intensity. Therefore, a 20 dB increase would represent a 100-fold increase in intensity.

Step 2: Use the Decibel Formula

The formula for converting sound intensity to decibels is given by \( L = 10 \times \log_{10}(\frac{I}{I_0}) \), where \( L \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity level (often the threshold of hearing, \( 10^{-12} \text{ W/m}^2 \)).

Step 3: Set Up Equations for Both Intensities

Given the sound levels, we have two equations from the decibel formula: 1. \( 106 = 10 \times \log_{10}(\frac{I_{\text{mower}}}{I_0}) \) for the power mower.2. \( 120 = 10 \times \log_{10}(\frac{I_{\text{concert}}}{I_0}) \) for the rock concert.

Step 4: Solve for intensities

Rearrange the formulas to solve for \(I\):1. \( \frac{I_{\text{mower}}}{I_0} = 10^{10.6} \) which gives \(I_{\text{mower}} = I_0 \times 10^{10.6} \).2. \( \frac{I_{\text{concert}}}{I_0} = 10^{12} \) which gives \(I_{\text{concert}} = I_0 \times 10^{12} \).

Step 5: Calculate the Intensity Ratio

The ratio of the intensities is given by:\[ \text{Ratio} = \frac{I_{\text{concert}}}{I_{\text{mower}}} = \frac{I_0 \times 10^{12}}{I_0 \times 10^{10.6}} \] Simplifying gives \( 10^{12 - 10.6} = 10^{1.4} \approx 25.12 \).

Key Concepts

Decibel Scale

The decibel scale is a fascinating and essential part of understanding sound measurement. It's a logarithmic scale, which means it measures quantities in a way that mirrors exponential growth. When you move up the decibel scale, each step represents a tenfold increase in sound intensity, not just a simple addition.
This means that when sound levels increase by 10 dB, the intensity becomes ten times stronger, making it a useful tool for comparing vastly different sound levels.
Using the decibel scale helps us make sense of the wide range of sounds we encounter, from a whisper at 30 dB to a rock concert at 120 dB. The logarithmic nature condenses this vast range into a more practical and comprehensible scale.

Sound Intensity

Sound intensity refers to the power carried by sound waves per unit area in a direction perpendicular to that area. It's a measure of how much sound energy passes through a certain area in one second.
Intensity is related to how we perceive loudness; higher intensity means louder sounds. However, the human ear doesn't hear sound intensity on a linear scale. That's why the decibel scale, which is logarithmic, fits our perception better.
The intensity of sound can vary greatly, which is why we use a reference intensity, typically the threshold of hearing at about \(10^{-12} \text{ W/m}^2\). This reference point is crucial for calculating decibels, allowing us to compare different sounds accurately.

Decibel Formula

To transform sound intensity into decibels, we use the decibel formula. It's a mathematical tool that converts the physical measurement of sound intensity into a more manageable number using logarithms.
The formula is:

• \( L = 10 \times \log_{10}\left(\frac{I}{I_0}\right) \)

Here, \( L \) represents the sound level in decibels, \( I \) the intensity of the sound, and \( I_0 \) the reference intensity level. This formula is fundamental for calculating decibel levels from measured sound intensities.
For instance, calculating the decibels from an intensity requires dividing the sound intensity by the reference intensity, taking the base 10 logarithm of the result, and then multiplying by 10. This gives us a decibel number that makes comparing different sound levels simpler and more intuitive.