Chapter 3.3, Problem 81E

To determine

Tofind:the difference in loudness between an vacuum cleaner with an intensity of ${10}^{-4}$ watt per square meter and a rustling leaves with an intensity of ${10}^{-11}$ watt per square meter.

Answer to Problem 81E

  $70\text{ dB}$

Explanation of Solution

Given:

The relationship between the number of decibels $\beta$ and the intensity of a sound I (in watts per square meter) is:

  $\beta =10\log \frac{I}{{10}^{-12}}=10\left(\log I+12\right)$

Concept Used:

Product property of logarithm states that for a positive real numbers a such that $a\ne 1$ :

  ${\log }_a\left(uv\right)={\log }_{a}u+{\log }_{a}v$

Power property of logarithm states that for a positive real numbers a such that $a\ne 1$ :

  ${\log }_a{u}^n=n{\log }_{a}u$

Quotient property of logarithm states that for a positive real numbers a such that $a\ne 1$ :

  ${\log }_a\left(\frac{u}{v}\right)={\log }_{a}u-{\log }_{a}v$

First find the number of decibels in vacuum cleaner with an intensity of ${10}^{-4}$ watt per square meter

Substitute $I={10}^{-4}$ in $\beta =10\left(\log I+12\right)$ ,

  ${\beta }_{cleaner}=10\left(\log \left({10}^{-4}\right)+12\right)$

Using the power property,

  ${\beta }_{cleaner}=10\left(-4\log \left(10\right)+12\right)$

Using ${\log }_{b}b=1$ ,

  $\begin{array}{c}{\beta }_{cleaner}=10\left(-4{\log }_{10}\left(10\right)+12\right)\\ =10\left(-4\left(1\right)+12\right)\\ =10\left(-4+12\right)\end{array}$

  $\begin{array}{c}=10\left(8\right)\\ =80\end{array}$

Now, find the number of decibels in rustling leaveswith an intensity of ${10}^{-11}$ watt per square meter

Substitute $I={10}^{-11}$ in $\beta =10\left(\log I+12\right)$ ,

  ${\beta }_{leaves}=10\left(\log \left({10}^{-11}\right)+12\right)$

Using the power property,

  ${\beta }_{leaves}=10\left(-11\log \left(10\right)+12\right)$

Using ${\log }_{b}b=1$ ,

  $\begin{array}{c}{\beta }_{leaves}=10\left(-11{\log }_{10}\left(10\right)+12\right)\\ =10\left(-11\left(1\right)+12\right)\\ =10\left(-11+12\right)\end{array}$

  $\begin{array}{c}=10\left(1\right)\\ =10\end{array}$

Now find the difference,

  $\begin{array}{c}{\beta }_{cleaner}-{\beta }_{leaves}=80-10\\ =70\text{ dB}\end{array}$

Conclusion:

Hence, the difference in loudness is approximately 70 dB.