To determine the intensity [tex]\( I \)[/tex] of the singer's voice if the sound level is given as 90 dB, we can use the formula:
[tex]\[ L = 10 \log \left(\frac{I}{I_0}\right) \][/tex]
where:
- [tex]\( L \)[/tex] is the loudness in decibels (dB),
- [tex]\( I \)[/tex] is the intensity of the sound in watts per square meter ([tex]\( \frac{W}{m^2} \)[/tex]),
- [tex]\( I_0 \)[/tex] is the reference intensity, given as [tex]\( 10^{-12} \frac{W}{m^2} \)[/tex].
Given:
- [tex]\( L = 90 \)[/tex] dB,
- [tex]\( I_0 = 10^{-12} \frac{W}{m^2} \)[/tex].
We need to solve for [tex]\( I \)[/tex]. Follow these steps:
1. Rewrite the formula to isolate [tex]\( \frac{I}{I_0} \)[/tex]:
[tex]\[ 90 = 10 \log \left(\frac{I}{I_0}\right) \][/tex]
2. Divide both sides by 10 to simplify:
[tex]\[ \frac{90}{10} = \log \left(\frac{I}{I_0}\right) \][/tex]
[tex]\[ 9 = \log \left(\frac{I}{I_0}\right) \][/tex]
3. Convert the logarithmic equation to its exponential form to isolate [tex]\( I \)[/tex]:
Recall that if [tex]\( \log_b(a) = c \)[/tex], then [tex]\( a = b^c \)[/tex]. Using this property:
[tex]\[ 10^9 = \frac{I}{I_0} \][/tex]
4. Multiply both sides by [tex]\( I_0 \)[/tex] to solve for [tex]\( I \)[/tex]:
[tex]\[ I = I_0 \cdot 10^9 \][/tex]
5. Substitute the given value for [tex]\( I_0 \)[/tex]:
[tex]\[ I = 10^{-12} \cdot 10^9 \][/tex]
6. Combine the exponents:
[tex]\[ I = 10^{-12 + 9} \][/tex]
[tex]\[ I = 10^{-3} \][/tex]
[tex]\[ I = 0.001 \,\frac{W}{m^2} \][/tex]
Thus, the intensity of the singer's voice when the sound level is 90 dB is:
[tex]\[ I = 0.001 \,\frac{W}{m^2} \][/tex]
