Chapter 8.4, Problem 37PPS

(a)

To determine

To Find: The number of decibels of the sound with the intensity of 1 watt per square meter.

Answer to Problem 37PPS

The number of decibels is $120$ .

Explanation of Solution

Given:

The given equation is $\beta =10{\log }_{10}\frac{1}{{10}^{-12}}$ .

The given table is shown in Table 1

Table 1

Sound	Intensity	Decibels
Pin Drop	${10}^0$	0
Normal Breathing	${10}^1$	1
Clothes dryer	${10}^6$	6
Subway train	${10}^{10}$	10
Firecracker	${10}^{12}$	12

Calculation:

Consider the decibels of the sound with the intensity of 1 watt per square meter is determined is,

  $\begin{array}{c}\beta =10{\log }_{10}\frac{1}{{10}^{-12}}\\ =10{\log }_{10}\left({10}^{12}\right)\\ =10\left(12\right)\\ =120\end{array}$

Thus, the number of decibels is $120$ .

(b)

To determine

To Find: The number of decimals for the sound with the intensity of ${10}^{-2}$ watts per square meter.

Answer to Problem 37PPS

The number of decibels of the sound is $100$ .

Explanation of Solution

Consider the decibels of the sound with the intensity of ${10}^{-2}$ watt per square meter is determined is,

  $\begin{array}{c}\beta =10{\log }_{10}\frac{{10}^{-2}}{{10}^{-12}}\\ =10{\log }_{10}\left({10}^{10}\right)\\ =10\left(10\right)\\ =100\end{array}$

(c)

To determine

To Find:

The reason for why the decibel of the sound is not 100 times as great.

Answer to Problem 37PPS

The decibels to change by 20.

Explanation of Solution

Given:

The number of decibels for the sound with the intensity of ${10}^{-2}$ watt per square meter.

Calculation:

Since, the power of the logarithm only changes by 2 so it is multiplied by10 before the logarithm every time thus, it is said that the decibels to change by 20.