Question

A person speaking normally produces a sound intensity of $40dB$ at a distance of $1m$. If the threshold intensity for reasonable audibility is $20dB$, the maximum distance at which he can be heard clearly is

• A. $4m$
• B. $5m$
• C. $10m$
• D. $20m$

Answer 1

Answer: (C)

$10m$

Loudness in decibels is given by: $\beta =10lo{g}_{10}\left(\frac{I}{I_0}\right)$
Where $I_0=$ threshold intensity of sound $={10}^{-12}W/m^2$
i.e., $40=10lo{g}_{10}\left(\frac{I_1}{I_0}\right)$ ................$\left(i\right)$
and $20=10lo{g}_{10}\left(\frac{I_2}{1_0}\right)$ ................$\left(ii\right)$
$\frac{\left(i\right)}{\left(ii\right)}\Rightarrow \frac{40}{20}=lo{g}_{10}\left(\frac{I_1}{I_2}\right)$
$2=lo{g}_{10}\left(\frac{I_1}{I_2}\right)$ or $\frac{I_1}{I_2}={10}^2$
$\therefore \frac{r_2^2}{r_1^2}={10}^2$ $($since $I\alpha \frac{1}{r^2})$
$r_2^2={10}^2{r}_1^2$ or $r_2=10r_1=10\times 1=10m$