Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 49 and \(50,\) find the Ievel of each sound \(\beta\). As a result of the installation of a muffler, the noise level of an engine was reduced from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise due to the installation of the muffler.

Step-by-Step Solution

Answer

The calculated intensity level before the muffler installation would be \( I_{1} = 10^{-11.2} W/m^2 \) and after the installation it would drop to \( I_{2} = 10^{-11.8} W/m^2 \). Running the numbers to compute the percentage decrease would yield a result about 74.08%.

Step 1: Calculate Initial Intensity

To begin with, using the given formula i.e. \( \beta = 10 \log _{10} \frac{I}{I_{0}} \), the initial intensity \( I_{1} \) could be calculated for the original noise level at 88 decibels. Rearrange the formula to get \( I_{1} = I_{0} * 10^{\frac{\beta_{1}}{10}} \) and substitute the given value i.e. \( I_{0} = 10^{-12} W/m^2 \) and \( \beta_{1} = 88 dB \).

Step 2: Calculate Final Intensity

Similarly, compute the final intensity \( I_{2} \) for the noise level at 72 decibels after the installation of the muffler. Use the same formula \( I_{2} = I_{0} * 10^{\frac{\beta_{2}}{10}} \) with \( I_{0} = 10^{-12} W/m^2 \) and \( \beta_{2} = 72 dB \).

Step 3: Calculate Decrease in Intensity

Now, to determine the percent decrease in the intensity level, firstly figure out the decrease in intensity \( \Delta I = I_{1} - I_{2} \) and then compute the percentage decrease using the formula \( \frac{\Delta I}{I_{1}} * 100\%\).

Key Concepts

Decibel Level Calculation

Understanding the concept of decibel level calculation is critical when dealing with sound intensity. The decibel (dB) is a logarithmic unit that quantifies the ratio of a particular sound intensity to a reference intensity. It is used because the human ear perceives sound intensity logarithmically.

The formula for calculating the level of sound in decibels is: \[\beta = 10 \log_{10} \frac{I}{I_{0}}\]where \(\beta\) is the sound level in decibels, \(I\) is the intensity of the sound in watts per square meter, and \(I_{0}\) is the reference intensity, typically set at \(10^{-12} W/m^2\), which represents the faintest sound that the average human ear can detect.

To calculate the sound level,

• Identify the sound's intensity.
• Divide the sound's intensity by the reference intensity \(I_{0}\).
• Take the logarithm of that ratio.
• Multiply by 10 to determine the sound level in decibels.

This formula demonstrates how a ten-fold increase in intensity increases the sound level by 10 decibels, reflecting a logarithmic perception of loudness.

Intensity Level Reduction

When a physical device like a muffler is introduced to a system to reduce noise, it leads to an intensity level reduction in the sound emitted. The aforementioned decibel level formula can be utilized to quantitatively assess this reduction.

To find the percent decrease in the intensity level due to the muffler:

• Compute the initial intensity level before the muffler was installed using the decibel formula.
• Compute the final intensity level after the muffler installation using the same formula.
• Calculate the difference in intensity \(\Delta I\) by subtracting the final intensity from the initial intensity.
• Finally, use the formula \(\frac{\Delta I}{I_{1}} \times 100\%\) to find the percentage decrease, where \(I_{1}\) is the initial intensity level.

This gives a clear idea of the effectiveness of the noise reduction device in terms of the percentage by which the intensity is lowered. Such an understanding is crucial, particularly in industrial settings where sound reduction can significantly impact worker health and environmental noise pollution.

Logarithmic Equations

Logarithmic equations play a pivotal role in calculating decibel levels and intensity reductions. A logarithm can be thought of as the inverse operation of exponentiation, and within the context of decibels, logarithms help to deal with the vast range of sound intensities.

For example, the decibel formula includes a logarithmic equation where one solves for sound intensity. To handle it effectively:

• Understand the properties of logarithms, such as the product, quotient, and power rules.
• Know how to convert logarithmic equations into exponential form to solve them more easily for the variable of interest.
• Be aware that in the context of sound, the base 10 logarithm is used, reflecting the fact that our ears perceive sound levels on a logarithmic scale.

Given the exponential nature of sound intensity, logarithmic equations enable the comparison of sound levels in a more manageable and understandable way. When you manipulate these equations correctly, you can calculate the change in intensity level due to an intervention, such as the introduction of a muffler, and express this change in a percentage that reflects real-world effects.