To calculate the loudness, [tex]\(L\)[/tex], of a sound intensity of [tex]\(l = 10^{-1}\)[/tex] watts per square meter using the formula [tex]\(L = 10 \log \frac{l}{I_0}\)[/tex], where [tex]\(I_0 = 10^{-12}\)[/tex] W/m[tex]\(^2\)[/tex] is the reference intensity level (the least intense sound a human ear can hear), we follow these steps:
1. Identify the given values:
- The sound intensity, [tex]\( l = 10^{-1} \)[/tex] W/m[tex]\(^2\)[/tex]
- The reference intensity level, [tex]\( I_0 = 10^{-12} \)[/tex] W/m[tex]\(^2\)[/tex]
2. Substitute the values into the formula:
[tex]\[
L = 10 \log \left( \frac{10^{-1}}{10^{-12}} \right)
\][/tex]
3. Simplify the fraction inside the logarithm:
[tex]\[
\frac{10^{-1}}{10^{-12}} = 10^{-1 - (-12)} = 10^{-1 + 12} = 10^{11}
\][/tex]
4. Take the logarithm base 10 of [tex]\(10^{11}\)[/tex]:
[tex]\[
\log (10^{11}) = 11
\][/tex]
5. Multiply by 10 to convert to decibels:
[tex]\[
L = 10 \times 11 = 110 \text{ dB}
\][/tex]
Therefore, the approximate loudness of a rock concert with a sound intensity of [tex]\(10^{-1}\)[/tex] W/m[tex]\(^2\)[/tex] is [tex]\(110 \text{ dB}\)[/tex].
The correct answer is:
[tex]\[ 110 \text{ dB} \][/tex]
