To solve this problem, let's follow a step-by-step approach using the provided formulas and given data:
1. Identify the given values:
- Intensity in a quiet room ([tex]\( I_{\text{quiet room}} \)[/tex]): [tex]\( 10^{-10} \)[/tex] watts/m²
- Threshold of hearing intensity ([tex]\( I_0 \)[/tex]): [tex]\( 10^{-12} \)[/tex] watts/m²
- The sound level increase on a busy street is 50 dB greater than in a quiet room.
2. Calculate the sound level in the quiet room ([tex]\( \beta_{\text{quiet room}} \)[/tex]):
The formula to calculate the decibel level ([tex]\( \beta \)[/tex]) is:
[tex]\[
\beta = 10 \log \left(\frac{I}{I_0}\right)
\][/tex]
Substitute [tex]\( I_{\text{quiet room}} \)[/tex] and [tex]\( I_0 \)[/tex] into the formula:
[tex]\[
\beta_{\text{quiet room}} = 10 \log \left(\frac{10^{-10}}{10^{-12}}\right)
\][/tex]
Calculate the ratio inside the logarithm:
[tex]\[
\frac{10^{-10}}{10^{-12}} = 10^2 = 100
\][/tex]
Now calculate the logarithm:
[tex]\[
\log(100) = 2
\][/tex]
Therefore:
[tex]\[
\beta_{\text{quiet room}} = 10 \times 2 = 20 \text{ dB}
\][/tex]
3. Calculate the sound level on a busy street ([tex]\( \beta_{\text{busy street}} \)[/tex]):
Given that the sound level on a busy street is 50 dB greater than in a quiet room:
[tex]\[
\beta_{\text{busy street}} = \beta_{\text{quiet room}} + 50 = 20 \text{ dB} + 50 \text{ dB} = 70 \text{ dB}
\][/tex]
4. Calculate the intensity of sound on the busy street ([tex]\( I_{\text{busy street}} \)[/tex]):
We use the inverse formula to find [tex]\( I \)[/tex] from [tex]\( \beta \)[/tex]:
[tex]\[
I = I_0 \times 10^{\frac{\beta}{10}}
\][/tex]
Substitute [tex]\( \beta_{\text{busy street}} \)[/tex] and [tex]\( I_0 \)[/tex] into the formula:
[tex]\[
I_{\text{busy street}} = 10^{-12} \times 10^{\frac{70}{10}}
\][/tex]
Simplify the exponent:
[tex]\[
10^{\frac{70}{10}} = 10^7
\][/tex]
Therefore:
[tex]\[
I_{\text{busy street}} = 10^{-12} \times 10^7 = 10^{-5} \text{ watts/m}^2
\][/tex]
5. Conclusively:
- The level of sound in the quiet room is [tex]\( 20 \)[/tex] dB.
- The intensity of sound on the busy street is [tex]\( 10^{-5} \)[/tex] watts/m².
Thus, the completed sentences with the correct values are:
- The level of sound in the quiet room is 20 dB.
- The intensity of sound in the busy street is [tex]\( 10^{-5} \)[/tex] watt/m².
