- Use the loudness formula \(L = 10 \log_{10} (\frac{I}{I_0})\) to compute the loudness in decibels for both music intensities.
- Jessica's loudness level is calculated as \(L_J = 10 \log_{10} (10^3) = 30 \text{ dB}\).
- Braylee's loudness level is calculated as \(L_B = 10 \log_{10} (10^9) = 90 \text{ dB}\).
- The ratio of Braylee's music loudness to Jessica's music loudness is \(\frac{90}{30} = \boxed{3}\).
Explanation
- Problem Analysis
The loudness of a sound, measured in decibels (dB), is given by the formula \(L = 10 \log_{10} \left(\frac{I}{I_0}\right)\), where \(I\) is the intensity of the sound in watts per square meter (\(W/m^2\)) and \(I_0 = 10^{-12} W/m^2\) is the reference intensity corresponding to the quietest sound a human ear can detect. We are given the sound intensity of music for two people: Jessica's music intensity \(I_J = 10^{-9} W/m^2\) and Braylee's music intensity \(I_B = 10^{-3} W/m^2\). Our goal is to determine how many times louder Braylee's music is than Jessica's, which corresponds to the ratio of their loudness levels in decibels, \(\frac{L_B}{L_J}\).
- Calculating Jessica's Loudness
First, we calculate the loudness of Jessica's music (\(L_J\)) in decibels using the formula:
\[L_J = 10 \log_{10} \left(\frac{I_J}{I_0}\right)\]
Substituting the known values \(I_J = 10^{-9}\) and \(I_0 = 10^{-12}\):
\[L_J = 10 \log_{10} \left(\frac{10^{-9}}{10^{-12}}\right)\]
Using exponent rules, \(\frac{10^{-9}}{10^{-12}} = 10^{-9 - (-12)} = 10^3\). Since \(\log_{10}(10^3) = 3\), we find:
\[L_J = 10 \times 3 = 30 \text{ dB}\]
- Calculating Braylee's Loudness
Next, we calculate the loudness of Braylee's music (\(L_B\)) in decibels using the same formula:
\[L_B = 10 \log_{10} \left(\frac{I_B}{I_0}\right)\]
Substituting the known values \(I_B = 10^{-3}\) and \(I_0 = 10^{-12}\):
\[L_B = 10 \log_{10} \left(\frac{10^{-3}}{10^{-12}}\right)\]
Using exponent rules, \(\frac{10^{-3}}{10^{-12}} = 10^{-3 - (-12)} = 10^9\). Since \(\log_{10}(10^9) = 9\), we find:
\[L_B = 10 \times 9 = 90 \text{ dB}\]
- Calculating the Ratio and Conclusion
Finally, we determine how many times louder Braylee's music is than Jessica's by calculating the ratio of their decibel levels:
\[\text{Ratio} = \frac{L_B}{L_J} = \frac{90}{30} = 3\]
Thus, Braylee's music is 3 times louder than Jessica's. We can conclude this corresponds to the second option provided. The final answer is \(\boxed{3}\).
Examples
Understanding the logarithmic scale for sound intensity is crucial in fields like environmental science and audio engineering. For example, when architects design a concert hall or when city planners assess traffic noise, they must understand how doubling the sound intensity relates to decibels (which is a logarithmic increase, not a linear one). This helps in setting safety standards for prolonged exposure to loud sounds, ensuring that sound levels remain within comfortable and safe ranges for humans.