To solve the given problem, we need to determine the loudness [tex]\( L \)[/tex], measured in decibels (dB), of a sound given its intensity [tex]\( i \)[/tex] using the formula:
[tex]\[ L = 10 \log_{10} \left( \frac{i}{i_0} \right) \][/tex]
where:
- [tex]\( i \)[/tex] is the sound intensity.
- [tex]\( i_0 \)[/tex] is the reference intensity, i.e., the least intense sound a human ear can hear, which is [tex]\( 10^{-12} \, \text{W/m}^2 \)[/tex].
In the problem, we are provided with the following values:
- [tex]\( i = 10^{-1} \, \text{W/m}^2 \)[/tex]
- [tex]\( i_0 = 10^{-12} \, \text{W/m}^2 \)[/tex]
Now, let's plug these values into the formula step-by-step:
1. Set up the formula with the given values:
[tex]\[ L = 10 \log_{10} \left( \frac{10^{-1}}{10^{-12}} \right) \][/tex]
2. Simplify the expression inside the logarithm:
[tex]\[ \frac{10^{-1}}{10^{-12}} = 10^{-1} \times 10^{12} = 10^{11} \][/tex]
3. Insert this result back into the formula:
[tex]\[ L = 10 \log_{10} (10^{11}) \][/tex]
4. Apply the property of logarithms:
Recall that [tex]\( \log_{10} (10^x) = x \)[/tex], where [tex]\( x \)[/tex] is the exponent.
[tex]\[ \log_{10} (10^{11}) = 11 \][/tex]
5. Now multiply by 10 to find the loudness in decibels:
[tex]\[ L = 10 \times 11 = 110 \, \text{dB} \][/tex]
So, the loudness of a rock concert with a sound intensity of [tex]\( 10^{-1} \, \text{W/m}^2 \)[/tex] is 110 decibels (dB).
Thus, the correct answer is:
[tex]\[ \boxed{110 \, \text{dB}} \][/tex]
