Problem 56 - Algebra 2

Question

New components reduce the sound intensity of a certain model of vacuum cleaner from \(10^{-4} \mathrm{W} / \mathrm{m}^{2}\) to \(6.31 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2} .\) By how many decibels do these new components reduce the vacuum cleaner's loudness?

Step-by-Step Solution

Answer

The new components reduce the vacuum cleaner's loudness by 22dB

Step 1: Calculate the initial loudness in decibels

Use the given initial sound intensity of \(10^{-4} \mathrm{W} / \mathrm{m}^{2}\), and the standard reference sound intensity \(I0 = 1.0 \times 10^{-12} \mathrm{W} / \mathrm{m}^{2}\), and insert into the formula for loudness: \(L1 = 10 * \log_{10}(I_1/I0) = 10 * \log_{10}(10^{-4}/10^{-12}) = 10 * 8 = 80 \, \text{dB}\)

Step 2: Calculate the final loudness in decibels

Use the given final sound intensity after the introduction of new components \(6.31 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\), and insert this into the formula for loudness to get the final decibel level: \(L2 = 10 * \log_{10}(I_2/I0) = 10 * \log_{10}(6.31 \times 10^{-6}/10^{-12})\approx 10 * 5.8 = 58 \, \text{dB}\)

Step 3: Find the difference

To find by how many decibels the new components reduce the vacuum cleaner's loudness, find the difference between the initial and final decibel levels: \(ΔL = L1 - L2 = 80dB - 58dB = 22dB\). So, the new components reduce the vacuum cleaner's loudness by 22 Decibels.

Key Concepts

Sound Intensity

Sound intensity is a measure of the power carried by sound waves per unit area in a direction perpendicular to that area. It is expressed in units of watts per square meter (W/m²). The concept of sound intensity helps us understand how loud or quiet a sound is.

In the context of everyday noises, like vacuum cleaners, sound intensity tells us how much sound energy is hitting your ear. Higher sound intensity means louder sounds, while lower sound intensity indicates quieter sounds. When discussing sound reduction, we often aim to bring down the sound intensity to make things quieter.

Understanding sound intensity is important when dealing with devices or environments that are sensitive to noise. By knowing how intense a sound is, engineers and designers can devise ways to reduce it. In our exercise, we saw a decrease from an intensity of \(10^{-4} \text{ W/m}^2\) to \(6.31 \times 10^{-6} \text{ W/m}^2\). This change directly influences how we perceive the loudness of the sound that the vacuum cleaner makes.

Logarithmic Function

Logarithmic functions play a crucial role in calculating the loudness of a sound. The formula used to convert sound intensity into a perceivable measure, like decibels, is based on logarithms. Decibels provide a way to express sound intensity in a scale that matches our human perception.

The formula for calculating loudness in decibels is:

• \(L = 10 \cdot \log_{10}(I/I_0)\)
  where \(L\) is the loudness in decibels, \(I\) is the sound intensity, and \(I_0\) is the reference sound intensity.

Logarithms turn multiplicative changes in sound intensity into additive changes in decibels. This property is very convenient because it means a steady increase in decibels reflects a consistent multiplication of intensity, which aligns well with how we hear things.
In our case, we applied the logarithmic formula to find that the initial loudness is 80 dB and the final is 58 dB, accurately reflecting the reduction in intensity due to new components.

Reference Sound Intensity

Reference sound intensity is an essential aspect when measuring sound levels in decibels. The standard reference sound intensity, \(I_0\), is defined as \(1.0 \times 10^{-12} \text{ W/m}^2\). This value is considered the quietest sound that the average human ear can hear.

Using a common reference intensity allows us to compare different sound levels on a consistent scale. It sets a baseline from which any other sound intensity can be compared and described in terms of decibels.

All decibel calculations made in acoustic science use this reference level. It effectively shifts our logarithmic scale so that 0 dB represents this threshold of hearing. Anything above this represents sounds louder than the quietest possible sound the human ear can detect.
In the reduction exercise, both the initial and reduced sound intensities were compared to this reference to determine their respective loudness in decibels. Without this context, it would be impossible to accurately convey just how much quieter the vacuum cleaner became after the modification.