# The intensity level of sound in a truck is 92 dB. What is the intensity of this sound?

Jun 5, 2017

$\approx 1.58 \times {10}^{-} 3 \frac{W}{m} ^ 2$

#### Explanation:

color(white)(-------)color(blue)[beta = 10log(I/I_o)

Where
$I = \text{sound intensity level (x)}$
${I}_{o} = \text{human threshold of hearing} \left(1 \times {10}^{-} 12 \frac{W}{m} ^ 2\right)$
$\beta = \text{sound intensity in dB (92 dB)}$

This is how the problem is setup

$\textcolor{w h i t e}{- - - -} 92 = 10 \log \left(\frac{x}{1 \times {10}^{-} 12 \frac{W}{m} ^ 2}\right)$

Now, we just solve for our sound intensity level $\left(x\right)$

$\textcolor{w h i t e}{- - - - - - -}$

Steps

• $92 = 10 \log \left(\frac{x}{1 \times {10}^{-} 12 \frac{W}{m} ^ 2}\right)$

• $\frac{92}{10} = \cancel{\frac{10}{10}} \log \left(\frac{x}{1 \times {10}^{-} 12 \frac{W}{m} ^ 2}\right)$

• $9.2 = \underbrace{\log} {\left(\frac{x}{1 \times {10}^{-} 12 \frac{W}{m} ^ 2}\right)}_{\textcolor{red}{\text{rewrite using log rules}}}$

• $9.2 = \log \left(x\right) - \log \left(1 \times {10}^{-} 12 \frac{W}{m} ^ 2\right)$

• 9.2 = log(x) - stackrel"-12"cancel[log(1xx10^-12 W/m^2)

• $9.2 = \log \left(x\right) + 12$

• $- 2.8 = \log \left(x\right)$

• ${10}^{-} 2.8 = x$

• color(magenta)[1.58 xx 10^-3 W/m^2] ~~ color(magenta)[x

$\textcolor{m a \ge n t a}{\text{Answer} : 1.58 \times {10}^{-} 3 \frac{W}{m} ^ 2} \approx \textcolor{m a \ge n t a}{x}$