To find the intensity of the sound of a jet plane given its sound level in decibels, we use the formula:
[tex]\[
\beta = 10 \log \left( \frac{I}{I_0} \right)
\][/tex]
where:
- [tex]\(\beta\)[/tex] is the sound level in decibels (dB),
- [tex]\(I\)[/tex] is the intensity of the sound in watts per square meter (W/m²),
- [tex]\(I_0\)[/tex] is the threshold of human hearing, which is [tex]\(10^{-12}\)[/tex] W/m².
We are given:
- [tex]\(\beta = 140 \, \text{dB}\)[/tex],
- [tex]\(I_0 = 10^{-12} \, \text{W/m}^2\)[/tex].
First, we rearrange the equation to solve for the intensity [tex]\(I\)[/tex]. The formula becomes:
[tex]\[
\frac{I}{I_0} = 10^{\frac{\beta}{10}}
\][/tex]
Next, we substitute the known values into the equation:
[tex]\[
\frac{I}{I_0} = 10^{\frac{140}{10}} = 10^{14}
\][/tex]
Since [tex]\(I_0 = 10^{-12} \, \text{W/m}^2\)[/tex], we replace [tex]\(I_0\)[/tex] in the rearranged formula:
[tex]\[
I = I_0 \cdot 10^{14} = 10^{-12} \cdot 10^{14}
\][/tex]
Simplifying the expression:
[tex]\[
I = 10^{14-12} = 10^2 = 100 \, \text{W/m}^2
\][/tex]
Hence, the intensity of the sound of a jet plane is approximately:
[tex]\[
\boxed{100}
\][/tex]
So, the correct answer is [tex]\( \text{100} \, \text{watts/meter}^2 \)[/tex].
