Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: \(10^{-10} \frac{W}{m^{2}},\) Vacuum: \(10^{-4} \frac{W}{m^{2}},\) Jet: \(10^{2} \frac{W}{m^{2}}\)

Step-by-Step Solution

Answer

Whisper: 20 dB, Vacuum: 80 dB, Jet: 140 dB

Step 1: Define the Logarithmic Formula

The intensity of a sound on a logarithmic scale is measured in decibels (dB). The formula to calculate the intensity in decibels is given by: \( I_{ ext{dB}} = 10 \cdot \log_{10}\left( \frac{I}{I_0} \right) \) where \( I \) is the intensity of the sound in \( \frac{W}{m^2} \), and \( I_0 = 10^{-12} \frac{W}{m^2} \) is the reference intensity.

Step 2: Calculate Decibels for Whisper

Substitute \( I = 10^{-10} \frac{W}{m^2} \) into the formula. \[ I_{ ext{dB}} = 10 \cdot \log_{10}\left( \frac{10^{-10}}{10^{-12}} \right) = 10 \cdot \log_{10} (10^2) = 10 \cdot 2 = 20 \] The intensity in decibels for a whisper is 20 dB.

Step 3: Calculate Decibels for Vacuum

Substitute \( I = 10^{-4} \frac{W}{m^2} \) into the formula.\[ I_{ ext{dB}} = 10 \cdot \log_{10}\left( \frac{10^{-4}}{10^{-12}} \right) = 10 \cdot \log_{10} (10^8) = 10 \cdot 8 = 80 \] The intensity in decibels for a vacuum cleaner is 80 dB.

Step 4: Calculate Decibels for Jet

Substitute \( I = 10^{2} \frac{W}{m^2} \) into the formula.\[ I_{ ext{dB}} = 10 \cdot \log_{10}\left( \frac{10^2}{10^{-12}} \right) = 10 \cdot \log_{10} (10^{14}) = 10 \cdot 14 = 140 \] The intensity in decibels for a jet is 140 dB.

Key Concepts

Intensity of Sound

Sound travels through air in waves and the intensity of these waves tells us how loud the sound is. Intensity is defined as the power per unit area carried by a wave and is measured in watts per square meter (W/m²). It is important to understand that the concept of intensity relates directly to how we perceive sound loudness.

Quiet sounds like whispers have low intensity, while louder sounds like a jet engine have high intensity:

• Whisper: Low intensity of around \(10^{-10} \frac{W}{m^2}\)
• Vacuum Cleaner: Moderate intensity, roughly \(10^{-4} \frac{W}{m^2}\)
• Jet: High intensity approaching \(10^{2} \frac{W}{m^2}\)

These examples show how different activities and tools can produce sound of varying intensities, affecting the way we hear them.

Decibels

Decibels (dB) are the units used to measure sound intensity level on a logarithmic scale. By using decibels, we can express this large range of intensities in a much more manageable way. The decibel scale is convenient because it is more aligned with the human ear's perception of sound loudness.

In general, an increase of 10 decibels represents roughly twice the perceived loudness of a sound. For instance:

• A whisper might be around 20 dB, which is quite soft.
• A vacuum cleaner can reach about 80 dB, significantly louder.
• A jet plane might reach around 140 dB, loud enough to cause hearing damage.

These examples illustrate how a relatively small change in decibels can reflect a large change in perceived sound intensity.

Logarithmic Formula

The logarithmic formula is used to convert sound intensity, measured in watts per square meter, to decibels. The formula is:\[I_{\text{dB}} = 10 \cdot \log_{10}\left( \frac{I}{I_0} \right)\]Here, \(I\) is the intensity of the sound, and \(I_0 = 10^{-12} \frac{W}{m^2}\) is a standard reference intensity.

This formula works by taking the logarithm (base 10) of the ratio between the sound intensity \(I\) and the reference intensity \(I_0\), then scaling it by 10. This scale factor of 10 is crucial as it makes the decibel scale match more closely with auditory perception, meaning that equal changes in decibels represent roughly equal changes in perceived loudness.