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The sound level, [tex]\beta[/tex], in decibels ( [tex]dB[/tex] ), of a sound with an intensity of [tex]I[/tex] is calculated in relation to the threshold of human hearing, [tex]I_0[/tex], by this equation:

[tex]\ \textless \ br/\ \textgreater \ \beta = 10 \log \left(\frac{I}{I_0}\right)\ \textless \ br/\ \textgreater \ [/tex]

The threshold of human hearing is [tex]10^{-12}[/tex] watts/meter [tex]^2[/tex]. The sound level of a jet plane is approximately [tex]140 \, dB[/tex]. The intensity of the sound of a jet plane is approximately [tex]\square[/tex] watts/meter [tex]^2[/tex].



Answer :

GinnyAnswer
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].

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