Chapter 5.4, Problem 127AYU

To determine

The loudness of normal conversation in decibels if the intensity of the sound is $x={10}^{-7}$ watt per square meter, where loudness $L\left(x\right)$ is given by $L\left(x\right)=10\log \frac{x}{I_0}$, where $x$ is the intensity of sound measured in watts per square meter, and $I_0={10}^{-12}$ watt per square meter is the least intense sound that human ear can detect.

Answer to Problem 127AYU

Solution:

The loudness of the normal conversation is $50\text{decibles}$.

Explanation of Solution

Given information:

The intensity of the sound is $x={10}^{-7}$ watt per square meter, loudness $L\left(x\right)=10\log \frac{x}{I_0}$ and $I_0={10}^{-12}$ watt per square meter.

Explanation:

Loudness is defined as $L\left(x\right)=10\log \frac{x}{I_0}$, when $x={10}^{-7}$, then the loudness of the normal conversation is $L\left({10}^{-7}\right)=10\log \frac{{10}^{-7}}{{10}^{-12}}$

$=10\log \left({10}^5\right)$

$=10\cdot 5$

$=50$

Therefore, the loudness of the normal conversation is $50$ decibels.