To solve for the sound intensity [tex]\( r \)[/tex] given that the sound level [tex]\( \beta \)[/tex] is 130 dB, we can use the formula [tex]\(\beta = 10 \log \left(\frac{r}{I_0}\right)\)[/tex], where [tex]\( I_0 \)[/tex] is the smallest sound intensity that can be heard by the human ear, approximately [tex]\( 10^{-12} \)[/tex] watts/meter[tex]\(^2\)[/tex].
Given:
[tex]\[
\beta = 130 \, \text{dB} \\
I_0 = 10^{-12} \, \text{watts/meter}^2
\][/tex]
Step-by-step solution:
1. Start with the formula:
[tex]\[
130 = 10 \log \left(\frac{r}{10^{-12}}\right)
\][/tex]
2. Divide both sides by 10 to isolate the logarithm:
[tex]\[
13 = \log \left(\frac{r}{10^{-12}}\right)
\][/tex]
3. Rewrite the logarithm equation in its exponential form:
[tex]\[
10^{13} = \frac{r}{10^{-12}}
\][/tex]
4. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = 10^{13} \times 10^{-12}
\][/tex]
5. Simplify the exponent:
[tex]\[
r = 10^{13-12} = 10^1 = 10
\][/tex]
Therefore, the sound intensity of a noise that is 130 dB is [tex]\( \boxed{10} \)[/tex] watts/meter[tex]\(^2\)[/tex].
