Chapter 3.5, Problem 48E

a.

To determine

Find the sound level using given data.

Answer to Problem 48E

The sound level is $-230$ decibels.

Explanation of Solution

Given:

The given model is $\beta =10\log \left(I{I}_0\right)$ , $I_0={10}^{-12}$ and the intensity for rustle of leaves is $I={10}^{-11}$ watt per ${\text{m}}^2$ .

Calculation:

Use $\beta =10\log \left(I{I}_0\right)$

Substitute the value of $I_0={10}^{-12}$ and $I={10}^{-11}$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^{-11}\times {10}^{-12}\right)\\ =10\log {10}^{-23}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-23\right)10\\ =-230\end{array}$

Hence the sound level is $-230$ decibels.

b.

To determine

Find the sound level using given data.

Answer to Problem 48E

The sound level is $-100$ decibels.

Explanation of Solution

Given:

The given model is $\beta =10\log \left(I{I}_0\right)$ , $I_0={10}^{-12}$ and the intensity for jet is $I={10}^2$ watt per ${\text{m}}^2$ .

Calculation:

Use $\beta =10\log \left(I{I}_0\right)$

Substitute the value of $I_0={10}^{-12}$ and $I={10}^2$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^2\times {10}^{-12}\right)\\ =10\log {10}^{-10}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-10\right)10\\ =-100\end{array}$

Hence the sound level is $-100$ decibels.

c.

To determine

Find the sound level using given data.

Answer to Problem 48E

The sound level is $-160$ decibels.

Explanation of Solution

Given:

The given model is $\beta =10\log \left(I{I}_0\right)$ , $I_0={10}^{-12}$ and the intensity for door slamming is $I={10}^{-4}$ watt per ${\text{m}}^2$ .

Calculation:

Use $\beta =10\log \left(I{I}_0\right)$

Substitute the value of $I_0={10}^{-12}$ and $I={10}^{-4}$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^{-4}\times {10}^{-12}\right)\\ =10\log {10}^{-16}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-16\right)10\\ =-160\end{array}$

Hence the sound level is $-160$ decibels.

d.

To determine

Find the sound level using given data.

Answer to Problem 48E

The sound level is $-140$ decibels.

Explanation of Solution

Given:

The given model is $\beta =10\log \left(I{I}_0\right)$ , $I_0={10}^{-12}$ and the intensity for siren is $I={10}^{-2}$ watt per ${\text{m}}^2$ .

Calculation:

Use $\beta =10\log \left(I{I}_0\right)$

Substitute the value of $I_0={10}^{-12}$ and $I={10}^{-2}$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^{-2}\times {10}^{-12}\right)\\ =10\log {10}^{-14}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-14\right)10\\ =-140\end{array}$

Hence the sound level is $-140$ decibels.