Q22E - University Physics with Modern Physics

Question

The Sacramento City Council adopted a law to reduce the allowed sound intensity level of the much-despised leaf blowers from their current level of about 95 dB to 70 db. With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

Step-by-Step Solution

Answer

The ratio of the new allowed intensity to the previously allowed intensity is $3.2\times {10}^{-3}$

STEP 1 Concept of the difference between the sound intensity levels for two different intensities

The formula is given as ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{-}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}\mathbf{10}\mathbf{log}\left(\frac{l_2}{l_1}\right)$ where, ${\mathbf{l}}_{\mathbf{2}}$ is the new intensity and ${\mathbf{l}}_{\mathbf{1}}$ is the previous intensity

STEP 2 Calculate the ratio of the new allowed intensity to the previously allowed intensity

New allowed intensity is ${\mathrm{l}}_2$ and the previously allowed intensity is ${\mathrm{l}}_1$, to find the ratio of these two use the equation ${\mathrm{\beta }}_2-{\mathrm{\beta }}_1=10\log \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right)$ and substitute the values in the equation we get,

$$\begin{gathered} \left(70\mathrm{db}-95\mathrm{db}\right)=10\log \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right) \\ -25\mathrm{db}=10\log \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right) \\ - 2.5=10\log \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right) \end{gathered}$$

 Note that ${10}^{\log \left(\mathrm{x}\right)}=\mathrm{x}$

Therefore, 

$$\begin{gathered} {10}^{-2.5}=\left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right) \\ \left(\frac{{\mathrm{l}}_2}{{\mathrm{l}}_1}\right)=3.2\times {10}^{-3} \end{gathered}$$

Thus, the ratio of the new allowed intensity to the previously allowed intensity is $3.2\times {10}^{-3}$