Question

Compare the intensity of sound at the pain level, \(120 \mathrm{~dB}\), with that at the whisper level, \(20 \mathrm{~dB}\).

Short answer

Answer: The intensity of sound at the pain level is \(10^{10}\) times more intense than the intensity of sound at the whisper level.

Step-by-step solution

Understanding Decibels and Intensity Formula

Decibels (dB) are used to express the intensity (power) of a sound wave. The formula for decibels is given by: \(I_\mathrm{dB} = 10\log_{10}\left(\frac{I}{I_0}\right)\) Where \(I_\mathrm{dB}\) is the intensity in decibels, \(I\) is the intensity of the sound wave, and \(I_0\) is the reference intensity (usually \(I_0 = 10^{-12} \mathrm{~W/m^2}\)). We need to find the intensity \(I\) for both pain level and whisper level using their respective decibel values, and then find the ratio between these intensities.

Convert dB to Intensity for Pain Level Sound

First, let's find the intensity of the pain level sound with \(I_{\mathrm{dB, pain}} = 120 \mathrm{~dB}\): \(120 = 10\log_{10}\left(\frac{I_\mathrm{pain}}{10^{-12}}\right)\) To find \(I_\mathrm{pain}\), we need to solve the equation for it. \(\frac{120}{10} = \log_{10}\left(\frac{I_\mathrm{pain}}{10^{-12}}\right)\) \(12 = \log_{10}\left(\frac{I_\mathrm{pain}}{10^{-12}}\right)\) Now, take the power of 10 for both sides of the equation: \(10^{12} = \frac{I_\mathrm{pain}}{10^{-12}}\) Finally, find \(I_\mathrm{pain}\): \(I_\mathrm{pain} = 10^{12} \cdot 10^{-12} = 1 \mathrm{~W/m^2}\)

Convert dB to Intensity for Whisper Level Sound

Now, let's find the intensity of the whisper level sound with \(I_{\mathrm{dB, whisper}} = 20 \mathrm{~dB}\): \(20 = 10 \log_{10}\left(\frac{I_\mathrm{whisper}}{10^{-12}}\right)\) Solve for \(I_\mathrm{whisper}\): \(\frac{20}{10} = \log_{10}\left(\frac{I_\mathrm{whisper}}{10^{-12}}\right)\) \(2 = \log_{10}\left(\frac{I_\mathrm{whisper}}{10^{-12}}\right)\) Take the power of 10 for both sides of the equation: \(10^2 = \frac{I_\mathrm{whisper}}{10^{-12}}\) Finally, find \(I_\mathrm{whisper}\): \(I_\mathrm{whisper} = 10^{2} \cdot 10^{-12} = 10^{-10} \mathrm{~W/m^2}\)

Find the Ratio of the Intensities

Now that we have the intensity for pain level sound (\(I_\mathrm{pain}\)) and whisper level sound (\(I_\mathrm{whisper}\)), let's find the ratio between these intensities: \(\frac{I_\mathrm{pain}}{I_\mathrm{whisper}} = \frac{1}{10^{-10}} = 10^{10}\) So, the intensity of sound at the pain level is \(10^{10}\) times more intense than the intensity of sound at the whisper level.