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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].
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].
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