Q15E - University Physics with Modern Physics

Question

Eavesdropping! You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds like only an average whisper of 20.0 db. How close should you move to the chatterboxes for the sound level to be 60.0 dB?

Step-by-Step Solution

Answer

The person is required to move 15.0 cm towards the direction of the conversation

STEP 1 Concept of Distance - intensity relation

To calculate the distance required to move by the person by using the distance-intensity relation is $\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\sqrt{\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}}$ were. $r_1$   is the initial distance. $r_2$ is the final distance. $l_2$ is the final intensity of the sound. $l_1$ is the initial intensity of the sound. 

STEP 2 Calculate the difference of two sound intensity level

The intensity of the sound at a distance 15.0 m apart is 20.0 dB.

Formula to calculate the difference of two sound intensity level is ${\mathrm{\beta }}_2-{\mathrm{\beta }}_1=10\log \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right)$ were, ${\beta }_2$ is the final intensity level of the sound. ${\beta }_1$ is the initial intensity level of the sound.

Substitute 60.0 dB for ${\beta }_2$ and 20.0 dB for ${\beta }_1$ in the above equation to find $\left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right)$

$$\begin{gathered} 60.0\mathrm{dB}=10\log \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right) \\ \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right)=40.0\mathrm{dB} \end{gathered}$$

STEP 3 CALCULATION.

Substitute 40.0dB for $\left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right)$ and 15.0m for $r_1$ to find $r_2$

$$\begin{gathered} \frac{15.0\mathrm{m}}{{\mathrm{r}}_2}=\sqrt{40.0\mathrm{dB}} \\ =15.0\mathrm{cm} \end{gathered}$$

Therefore, the person is required to move 15.0 cm towards the direction of the conversation