Chapter 3.5, Problem 47E

a.

To determine

Find the sound level using given data.

Answer to Problem 47E

The sound level is $-220$ 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 quiet room is $I={10}^{-10}$ watt per ${\text{m}}^2$ .

Calculation:

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

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

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^{-10}\times {10}^{-12}\right)\\ =10\log {10}^{-22}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-22\right)10\\ =-220\end{array}$

Hence the sound level is $-220$ decibels.

b.

To determine

Find the sound level using given data.

Answer to Problem 47E

The sound level is $-170$ 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 street corner is $I={10}^{-5}$ watt per ${\text{m}}^2$ .

Calculation:

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

Substitute the value of $I_0={10}^{-12}$ and $I={10}^{-5}$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^{-5}\times {10}^{-12}\right)\\ =10\log {10}^{-17}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-17\right)10\\ =-170\end{array}$

Hence the sound level is $-170$ decibels.

c.

To determine

Find the sound level using given data.

Answer to Problem 47E

The sound level is $-200$ 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 quiet radio is $I={10}^{-8}$ watt per ${\text{m}}^2$ .

Calculation:

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

Substitute the value of $I_0={10}^{-12}$ and $I={10}^{-8}$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^{-8}\times {10}^{-12}\right)\\ =10\log {10}^{-20}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-20\right)10\\ =-200\end{array}$

Hence the sound level is $-200$ decibels.

d.

To determine

Find the sound level using given data.

Answer to Problem 47E

The sound level is $-120$ 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 threshold of pain is $I={10}^0$ watt per ${\text{m}}^2$ .

Calculation:

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

Substitute the value of $I_0={10}^{-12}$ and $I={10}^0$ in the given equation.

  $\begin{array}{l}\Rightarrow \beta =10\log \left({10}^0\times {10}^{-12}\right)\\ =10\log {10}^{-12}\end{array}$

Use power rule.

  $\begin{array}{l}=\left(-12\right)10\\ =-120\end{array}$

Hence the sound level is $-120$ decibels.