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The loudness, $L$, measured in decibels (dB), of a sound intensity, $I$, measured in watts per square meter, is defined as $L=10 \log \frac{I}{I_0}$, where $I_0=10^{-12}$ and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is $10^{-10}$. Ahmad, Brandon's neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is $10^{-4}$. How does Brandon's sound intensity level compare to Ahmad's mower?


ABrandon's sound intensity is $\frac{1}{4}$ the level of Ahmad's mower.
BBrandon's sound intensity is $\frac{1}{6}$ the level of Ahmad's mower.
CBrandon's sound intensity is 20 times the level of Ahmad's mower.
DBrandon's sound intensity is 80 times the level of Ahmad's mower.

Verified Answer :

  • Calculate the sound intensity level for Brandon as \(L_B = 10 \log_{10} (\frac{10^{-10}}{10^{-12}}) = 20 \text{ dB}\).
  • Calculate the sound intensity level for Ahmad's mower as \(L_A = 10 \log_{10} (\frac{10^{-4}}{10^{-12}}) = 80 \text{ dB}\).
  • Compare the two levels by finding the ratio: \(\frac{L_B}{L_A} = \frac{20}{80} = \frac{1}{4}\).
  • Conclude that Brandon's sound intensity level is \(\boxed{\frac{1}{4}}\) the level of Ahmad's mower.

Explanation

  1. Understanding Sound Intensity and Loudness

To determine how Brandon's sound intensity level compares to that of Ahmad's mower, we first need to understand the definitions of intensity and intensity level (loudness). The sound intensity level \(L\), measured in decibels (dB), is calculated using the formula \[L = 10 \log_{10} \frac{I}{I_0}\] where \(I\) is the sound intensity in \(\text{W/m}^2\) and \(I_0 = 10^{-12} \text{ W/m}^2\) is the reference intensity corresponding to the threshold of human hearing.

We are given:

  • Brandon's sound intensity: \(I_B = 10^{-10} \text{ W/m}^2\)
  • Ahmad's mower sound intensity: \(I_A = 10^{-4} \text{ W/m}^2\)
  1. Calculating Sound Intensity Levels (Loudness)

Now we calculate the sound intensity levels (loudness) for both Brandon and Ahmad's mower using the given formula:

  • For Brandon:

\[L_B = 10 \log_{10} \left( \frac{10^{-10}}{10^{-12}} \right) = 10 \log_{10} (10^2) = 10 \times 2 = 20 \text{ dB}\]

  • For Ahmad's mower:

\[L_A = 10 \log_{10} \left( \frac{10^{-4}}{10^{-12}} \right) = 10 \log_{10} (10^8) = 10 \times 8 = 80 \text{ dB}\]

  1. Comparing the Sound Levels

We compare the two sound intensity levels by looking at their ratio:
\[\frac{L_B}{L_A} = \frac{20 \text{ dB}}{80 \text{ dB}} = \frac{1}{4}\]
This means that Brandon's sound intensity level is \(\frac{1}{4}\) the level of Ahmad's mower. Comparing this to the given options, we find that the first choice correctly describes the relationship between the sound intensity levels calculated.

Examples

Understanding sound levels is crucial in fields like environmental science and urban planning. For example, local governments use decibel measurements to establish noise ordinances, ensuring that construction or commercial activity does not disturb residential areas. By calculating the ratio between different sound sources, as we did here, authorities can quantitatively assess how much louder one environment is compared to another, helping them maintain a healthy acoustic environment for the community.

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