- Identify the formula for loudness: \(L = 10 \log_{10} \left(\frac{I}{I_0}\right)\).
- Substitute the given intensities \(I = 10^{-1}\) and \(I_0 = 10^{-12}\) into the formula.
- Simplify the ratio to \(10^{11}\) and evaluate the logarithm to find \(L = 10 \times 11\).
- State the final result: \(\boxed{110 \text{ Db}}\).
Explanation
- Analysis of the Problem
The loudness \(L\) of a sound is measured in decibels (Db) and is determined by the formula \(L = 10 \log_{10} \left(\frac{I}{I_0}\right)\). In this problem, the reference sound intensity is \(I_0 = 10^{-12}\) watts/m², representing the threshold of human hearing, and the given sound intensity of the rock concert is \(I = 10^{-1}\) watts/m². We aim to find the value of \(L\) using these values.
- Calculating the Loudness
To calculate the loudness \(L\), we substitute the given values into the formula:
\[L = 10 \log_{10} \left(\frac{10^{-1}}{10^{-12}}\right)\]
First, simplify the fraction inside the logarithm by using the laws of exponents:
\[\frac{10^{-1}}{10^{-12}} = 10^{-1 - (-12)} = 10^{-1 + 12} = 10^{11}\]
Now, the formula becomes:
\[L = 10 \log_{10} (10^{11})\]
Using the property of logarithms \(\log_{b}(b^n) = n\), we have \(\log_{10} (10^{11}) = 11\). Thus, the loudness is:
\[L = 10 \times 11 = 110 \text{ Db}\]
- Conclusion and Final Answer
Comparing the calculated result to the given options, we find that the loudness of the rock concert is \(110\) Db. Therefore, the correct option is \(110\) Db. The final answer is \(\boxed{110 \text{ Db}}\).
Examples
Loudness is commonly used in audio engineering to manage sound levels. For instance, knowing the decibel levels of different environments helps in choosing the right type of protective equipment. If you are attending a rock concert, realizing it reaches 110 decibels helps you understand why you might need earplugs to prevent long-term damage to your hearing. Understanding these logarithmic scales allows for better health and safety decisions in loud environments.