Question

During takeoff, the sound intensity level of a jet engine is 140 dB at a distance of 30 m. What is the sound intensity level at a distance of 1.0 km?

Short answer

The sound intensity level at a distance of$1.0km$is$110dB$.

Step-by-step solution

Given information

Intensity level of a jet engine is $\beta =140dB$

Distance$1.0km=1000m$

Sound intensity at 1.0 km

The sound intensity at $1.0km$is

$$\begin{gathered} I\text{'}=\left(\frac{30m}{1000m}\right)I \\ I\text{'}=\left(\frac{3}{100}\right)I \end{gathered}$$

Sound intensity level at 1.0 km

The sound intensity level is given by $\beta =\left(10dB\right){\log }_{10}\left(\frac{I\text{'}}{I_0}\right)$.

Substitute the given values

$$\begin{gathered} \beta \text{'}=\left(10dB\right){\log }_{10}\left(\frac{{\left({\displaystyle \frac{3}{100}}\right)}^{2}I}{I_0}\right) \\ \beta \text{'}=\left(10dB\right){\log }_{10}{\left(\frac{3}{100}\right)}^2+\left(10dB\right){\log }_{10}\left(\frac{I}{I_0}\right) \\ \beta \text{'}=-30db+140dB \\ \beta \text{'}=110dB \end{gathered}$$

Therefore, the sound intensity level at a distance of$1.0km$is$110dB$.