You and a friend are watching fireworks that are launching from Toledo Volleyball Court. You are standing at Hensville Park (480 m) and your friend is at the Maritime Academy of Toledo (702 m). The sound intensity at the Maritime Academy is 1 W/m2.
What is the sound intensity level at your location?
To find the sound intensity level at your location, you can use the formula:
\[
\text{Sound Intensity Level (dB)} = 10 \times \log_{10}\left(\frac{I}{I_0}\right)
\]
where:
- \(I\) = intensity of the sound at your location
- \(I_0\) = reference intensity (usually set at \(10^{-12}\) W/m²)
First, calculate the sound intensity at your location using the inverse square law, which states that sound intensity decreases with the square of the distance from the source:
\[
\frac{I_{\text{Maritime Academy}}}{I_{\text{Hensville Park}}} = \left(\frac{r_{\text{Hensville Park}}}{r_{\text{Maritime Academy}}}\right)^2
\]
\[
\frac{1\, \text{W/m²}}{I_{\text{Hensville Park}}} = \left(\frac{702\, \text{m}}{480\, \text{m}}\right)^2
\]
\[
I_{\text{Hensville Park}} = \frac{1\, \text{W/m²}}{\left(\frac{702}{480}\right)^2} = \frac{1\, \text{W/m²}}{2.03125} \approx 0.4921\, \text{W/m²}
\]
Now, plug in the values into the formula:
\[
\text{Sound Intensity Level (dB)} = 10 \times \log_{10}\left(\frac{0.4921}{10^{-12}}\right)
\]
\[
\text{Sound Intensity Level (dB)} = 10 \times \log_{10}\left(4.921 \times 10^{11}\right)
\]
\[
\text{Sound Intensity Level (dB)} = 10 \times 11.693
\]
\[
\text{Sound Intensity Level (dB)} \approx 116.93\, \text{dB}
\]
Therefore, the sound intensity level at your location at Hensville Park is about 116.93 dB.