Chapter 3.3, Problem 87E

To determine

To calculate: The intensity of sound in decibel for the given scenario.

Answer to Problem 87E

  $70\text{ decibel}$ .

Explanation of Solution

Given information:

The number of decibel and intensity of sound is given by −

  $\beta =10\log \left(\frac{I}{{10}^{-12}}\right)$ where $\beta$ is number of decibel and $I$ is intensity of sound in watt per square meter.

Intensity of a vacuum cleaner is ${10}^{-4}$ watt per square meter.

Intensity of rustling leaves is ${10}^{-11}$ watt per square meter.

Calculation:

Intensity of a vacuum cleaner is given as ${10}^{-4}$ watt per square meter.

So, the number of decibel will be given as-

  $\begin{array}{l}\beta =10\log \left(\frac{I}{{10}^{-12}}\right)\\ \Rightarrow \beta =10\log \left(\frac{{10}^{-4}}{{10}^{-12}}\right)\\ \Rightarrow \beta =10\log {10}^8\\ \Rightarrow \beta =10\times 8\times 1\left\{\text{Since, }\log 10=1\right\}\\ \Rightarrow \beta =80\text{ decibel}\end{array}$

Intensity of rustling leaves is ${10}^{-11}$ watt per square meter.

Similarly, the number of decibel will be given as-

  $\begin{array}{l}\beta =10\log \left(\frac{I}{{10}^{-12}}\right)\\ \Rightarrow \beta =10\log \left(\frac{{10}^{-11}}{{10}^{-12}}\right)\\ \Rightarrow \beta =10\log 10\\ \Rightarrow \beta =10\times 1\left\{\text{Since, }\log 10=1\right\}\\ \Rightarrow \beta =10\text{ decibel}\end{array}$

Hence, difference in intensity of sound in decibel is $\left(80-10\right)=70\text{ decibel}$ .