Question

The Sacramento City Council adopted a law to reduce the allowed sound intensity level of the much-despised leaf blowers from their current level of about 95 dB to 70 dB. With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

Short answer

The new intensity is about 0.00316 times the previous intensity.

Step-by-step solution

Understand Decibels and Intensity

Decibels (dB) is a logarithmic scale used to measure the intensity of sound. The intensity level in dB is given by the formula: \( L = 10 \cdot \log_{10}(\frac{I}{I_0}) \), where \( L \) is the sound level in dB, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity (usually \( 10^{-12} \text{ W/m}^2 \)).

Set Up the Intensity Equations

For the initial intensity level of 95 dB, we have the equation: \( 95 = 10 \cdot \log_{10}(\frac{I_1}{I_0}) \). For the new intensity level of 70 dB, the equation is: \( 70 = 10 \cdot \log_{10}(\frac{I_2}{I_0}) \).

Solve for Initial Intensity

From \( 95 = 10 \cdot \log_{10}(\frac{I_1}{I_0}) \), solve for \( I_1 \):\[ \log_{10}(\frac{I_1}{I_0}) = 9.5 \]Then, \( \frac{I_1}{I_0} = 10^{9.5} \), so \( I_1 = 10^{9.5} \cdot I_0 \).

Solve for New Intensity

From \( 70 = 10 \cdot \log_{10}(\frac{I_2}{I_0}) \), solve for \( I_2 \):\[ \log_{10}(\frac{I_2}{I_0}) = 7 \]Then, \( \frac{I_2}{I_0} = 10^{7} \), so \( I_2 = 10^7 \cdot I_0 \).

Calculate the Intensity Ratio

The ratio of the new intensity to the previous intensity is \( \frac{I_2}{I_1} = \frac{10^7 \cdot I_0}{10^{9.5} \cdot I_0} = \frac{10^7}{10^{9.5}} = 10^{-2.5} \). Calculating \( 10^{-2.5} \) gives approximately 0.00316.

Key concepts

Decibels

Decibels (abbreviated as dB) are a way to describe sound intensity. Unlike most scales, which are linear, decibels are a **logarithmic scale**. This means that each increase of 10 dB represents a tenfold increase in sound intensity.
So if a sound is 70 dB, it’s not just **35 dB quieter** than a 105 dB sound but many times quieter in actual intensity.
Using decibels helps manage and compare sound levels in a way that aligns with human hearing. Humans perceive a change in 10 dB as about twice as loud or half as quiet, making decibels very useful in practical applications like measuring noise pollution or setting safe listening levels. The relevant formula for calculating decibels is:\[L = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right)\]Here,- **L** is the sound level in decibels.- **I** is the sound intensity.- **I_0** is the reference intensity, usually considered as \(10^{-12} \text{ W/m}^2\).Understandably, decibels make working with very large or small numbers easier, transforming them into more manageable figures. When laws such as the Sacramento City Council’s change permitted sound levels, they use dB for direct communication thereof.

Logarithmic Scale

A logarithmic scale is a type of scale used for a large range of quantities. When dealing with sound intensity, a logarithmic scale helps illustrate the exponential growth of sound waves effectively.
**How Does It Work?**
- A logarithmic scale measures quantities in powers of 10. Every unit increase on a logarithmic scale correlates to a tenfold increase in the corresponding linear scale. - It offers a way to compress wide-ranging data values into an understandable format. This is crucial when detecting changes in sound levels because human hearing perceives increase on a logarithmic rather than linear basis. **Why Is It Important?**
- In sound measurements, using a linear scale would result in a cumbersome comparison due to the wide variety in sound intensity values. For example, ordinary speech has an intensity that can be millions of times more than the faintest sound the human ear can detect. A familiarity with the logarithmic scale equips us with a better understanding and finer resolution when evaluating the real-world impact of transformations in sound intensities, like the change from 95 dB to 70 dB in Sacramento.

Sound Level Equations

Equations play a vital role in determining exact sound levels and comparing different sound intensities. The decibel level, as calculated by the sound level equation, provides a meaningful number to compare and analyze sound intensity.
**Understanding the Equations**: The sound level equation \(L = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right)\) allows us to find the sound level in decibels. For example, with a sound intensity change, like in Sacramento’s leaf blower law, we want to compare how much quieter one setting is compared to another.
- Original Intensity Level (95 dB) Equation: \[95 = 10 \cdot \log_{10}\left(\frac{I_1}{I_0}\right)\]- New Intensity Level (70 dB) Equation: \[70 = 10 \cdot \log_{10}\left(\frac{I_2}{I_0}\right)\]**Solving the Equations**:
To find the actual intensity ratio, change equations around to deduce:- Initial intensity calculation: \(I_1 = 10^{9.5} \cdot I_0\)- New intensity calculation: \(I_2 = 10^7 \cdot I_0\)By solving the difference, you find the intensity ratio of the new limit to the old limit. Calculations yield a ratio of about 0.00316 which shows the new sound level is significantly lower.
These equations underscore the power of sound level analysis in policymaking and public welfare scenarios.