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The formula [tex]\beta=10 \log \left(\frac{I}{I_0}\right)[/tex] is used to find the sound level, [tex]\beta[/tex], in decibels (dB), of a sound with an intensity of [tex]I[/tex]. In the formula, [tex]I_0[/tex] represents the smallest sound intensity that can be heard by the human ear (approximately [tex]10^{-12}[/tex] watts/meter [tex]^2[/tex]).

What is the sound intensity of a noise that is 130 dB?

The sound intensity is [tex]\square[/tex] watts/meter [tex]^2[/tex].



Answer :

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].