Problem 32 - Algebra 2

Question

A sound barrier along a highway reduced the intensity of the noise reaching a community by 95\(\% .\) By how many decibels was the noise reduced?

Step-by-Step Solution

Answer

Solving the final equation \(ΔDB = 10\cdot log_{10}(1/0.05)\) gives \(ΔDB = 13\) decibels. Hence, the intensity of the noise was reduced by approximately 13 decibels.

Step 1: Calculate the original and final intensities

To find the reduction in decibels, it is necessary, to begin with, calculating the original and final intensities of the noise. Since the percentage reduction in intensity is given, it can be inferred that if the original intensity is considered 100% then a 95% reduction means the final intensity is 5% of the original intensity.

Step 2: Substitute these values in decibel formula

Now, these rhertorical original and final intensities shall be replaced in the decibel formula mentioned in the analysis.\nLet \(I_{1}\) represent the initial intensity and \(I_{2}\) the final one: \(DB_{1} = 10\cdot log_{10}(I_{1}/I_{0})\) and \(DB_{2} = 10\cdot log_{10}(I_{2}/I_{0})\)

Step 3: Calculate difference in decibels

Now, the change (reduction) in decibels is the difference in the dB levels before and after the noise reduction. Consequently, \(ΔDB = DB_{1} - DB_{2} = 10\cdot log_{10}(I_{1}/I_{0}) - 10\cdot log_{10}(I_{2}/I_{0})\).\nThis can be simplified further using the properties of log to: \(ΔDB = 10\cdot [log_{10}(I_{1}/I_{0}) - log_{10}(I_{2}/I_{0})] = 10\cdot log_{10}((I_{1}/I_{2})\).\nGiven that \(I_{2}=0.05\cdot I_{1}\), the equation becomes \(ΔDB = 10\cdot log_{10}(1/0.05)\)

Step 4: Simplify for Answer

Solving the final equation will give the value of \(ΔDB\), the amount in decibels that the sound intensity was reduced.

Key Concepts

Sound Intensity

Sound intensity is a measure of how much sound energy passes through a certain area. It tells us how powerful a sound is, which affects how loud it seems. The intensity is often measured in watts per square meter (W/m²). A higher sound intensity means a louder sound, while a lower sound intensity means a quieter sound.

When we talk about changing sound intensity, like in the case of the highway noise reduction, we are reducing the sound energy that reaches the listener. In the given exercise, the sound intensity reaching the community was reduced by 95%. This means that the sound energy was almost completely blocked, resulting in a much quieter environment.

Sound intensity is crucial in understanding how environmental factors like barriers can alter sound levels to make them more pleasant or safe for communities.

Noise Reduction

Noise reduction involves decreasing the amount of unwanted sound in an environment. It is a crucial aspect of acoustics, helping to create a less intrusive and more comfortable space.

In the example of the highway noise barrier, noise reduction is achieved by cutting down the sound intensity that reaches the community. This is typically done by:

• Using physical barriers that absorb or reflect sound to prevent it from reaching a certain point.
• Implementing materials that block or diffuse sound energy, preventing large amounts from passing through.

Effective noise reduction can lead to a significant improvement in the quality of life for those affected by loud environments. By lowering sound intensity, it makes the area quieter and more conducive to activities like conversation and relaxation.

Logarithmic Scale

A logarithmic scale is used in measuring quantities that span a large range. In acoustics, loudness is measured using a logarithmic scale because sounds can vary greatly in intensity.

The decibel (dB) scale is a logarithmic scale used to express sound levels. It's based on powers of ten. This means each increase of 10 decibels represents a tenfold increase in sound intensity.

For example, if the original intensity is described in decibels as \(DB_1\), a reduction creates a new decibel level \(DB_2\). The difference \(ΔDB\), shows how much the intensity has decreased. This is calculated using the formula:

• \(ΔDB = 10 \cdot \log_{10}(I_1/I_2)\)

In the given scenario, the sound intensity was reduced to 5% of its original intensity. This translates to a significant drop in decibels, making it much more manageable by the community.
Understanding logarithmic scales in sound shows why even a small reduction in sound intensity can lead to a noticeable drop in perceived loudness.