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Chapter 4: Problem 97

Noise Pollution The level of a sound in decibels (db) is determined by the formula $$\text { sound level }=10 \cdot \log \left(I \times 10^{12}\right) \mathrm{db}$$ where \(I\) is the intensity of the sound in watts per square meter. To combat noise pollution, a city has an ordinance prohibiting sounds above 90 db on a city street. What value of \(I\) gives a sound of 90 db?

Short Answer

Expert verified
I = 10^{-3} watts per square meter

Step by step solution

01

Write Down the Given Formula

The formula to determine the sound level in decibels (\text{db}) is given as: \[ \text{sound level} = 10 \cdot \log(I \times 10^{12}) \text{ db} \]
02

Substitute the Given Sound Level

Substitute the given sound level of 90 db into the equation: \[ 90 = 10 \cdot \log(I \times 10^{12}) \]
03

Isolate the Logarithmic Expression

Divide both sides of the equation by 10 to isolate the logarithmic expression: \[ 9 = \log(I \times 10^{12}) \]
04

Remove the Logarithm Using Exponentiation

Rewrite the logarithmic equation in its exponential form to solve for \(I\). Recall that \( \log_b(a) = c \) implies \( b^c = a \): \[ 10^9 = I \times 10^{12} \]
05

Solve for the Intensity I

Isolate \(I\) by dividing both sides of the equation by \(10^{12}\): \[ I = \frac{10^9}{10^{12}} = 10^{-3} \text{ watts per square meter} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithmic functions
Logarithmic functions are essential in various scientific fields, especially when dealing with large or small numbers. A logarithm can be thought of as the inverse of an exponential function. For instance, if you have a number, say 1000, and you want to find out what power of 10 gives you 1000, you would use a logarithm. The logarithm base 10 of 1000 is 3, written as \(\text{log}_{10}(1000) = 3\). This tells you that 10 raised to the power of 3 equals 1000.
Logarithms are used in the given sound level formula essentially to compress large ranges of values into smaller ranges. The logarithmic scale helps in managing these vast ranges of sound intensities in a compact and comprehensible way.
In the problem, we use the logarithmic function to find the intensity of a sound that corresponds to a certain decibel level. By transforming the logarithmic equation into an exponential form, we solve for the sound intensity in a straightforward manner.
sound intensity
Sound intensity is a measure of the energy a sound wave carries per unit area. It is usually measured in watts per square meter (W/m\(^2\)). The higher the intensity, the louder the sound.
The decibel (dB) scale is a logarithmic scale used to describe the intensity of sound. This scale is convenient because the human ear perceives sound intensity logarithmically rather than linearly. Thus, each increase of 10 dB corresponds roughly to a doubling of perceived loudness, even though the actual power changes by a factor of 10.
In the given exercise, we see that the formula for sound level uses the base-10 logarithm of the intensity multiplied by 10\(^12\). This adjustment (multiplying by 10\(^12\)) helps bring typical sound intensities into a more manageable range for calculation and discussion.
noise pollution
Noise pollution refers to harmful or annoying levels of noise such as those from industrial activity, traffic, or urban construction. High noise levels can have various adverse effects on both humans and wildlife, including hearing loss, stress, and sleep disturbances.
Municipal regulations often set limits on acceptable sound levels to combat this issue, usually measured in decibels. In the given problem, the city has set a maximum allowable noise level of 90 dB for city streets, beyond which sound is considered too loud and potentially harmful.
The problem requires calculating the corresponding sound intensity for a noise level of 90 dB. This aids in setting equipment and practices that comply with noise regulations, helping to minimize noise pollution and its detrimental effects.

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Most popular questions from this chapter

Two-Parent Families The percentage of households with children that consist of two-parent families is shown in the following table (U.S. Census Bureau, www.census.gov). $$\begin{array}{|c|c|} \hline \text { Year } & \text { Two Parents } \\ \hline 1970 & 85 \% \\ 1980 & 77 \\ 1990 & 73 \\ 1992 & 71 \\ 1994 & 69 \\ 1996 & 68 \\ 1998 & 68 \\ 2000 & 68 \\ \hline \end{array}$$ a. Use logarithmic regression on a graphing calculator to find the best- fitting curve of the form \(y=a+b \cdot \ln (x)\) where \(x=0\) corresponds to 1960. b. Use your equation to predict the percentage of two-parent families in 2010 . c. In what year will the percentage of two-parent families reach \(50 \% ?\). d. Graph your equation and the data on your graphing calculator. Does this logarithmic model look like another model that we have used?

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