/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 Use the following information fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 68

Use the following information for determining sound intensity. The level of sound \(\boldsymbol{\beta}\), in decibels, with an intensity of \(I\), is given by \(\boldsymbol{\beta}=10 \log \left(I / I_{0}\right),\) 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 65 and 66 , find the level of sound \(\boldsymbol{\beta}\). Due to 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 as a result of the installation of the muffler.

Short Answer

Expert verified
The percentage decrease in the intensity of the noise as a result of the installation of the muffler is the result obtained in Step 3

Step by step solution

01

Find the intensity before muffler installation

The sound intensity \( I_{1} \) before the muffler was installed can be calculated using the formula \( \beta = 10 \log(I / I_{0}) \), with \( \beta = 88 \) decibels and \( I_{0} = 10^{-12} \). Re-arrange this equation to \( I_{1} = I_{0} * 10^(\beta / 10) \), and fill in the given values.
02

Find the intensity after muffler installation

Similarly, the sound intensity \( I_{2} \) after the muffler was installed can be calculated using the same formula, but now with \( \beta = 72 \) decibels. Re-arrange the equation to \( I_{2} = I_{0} * 10^(\beta / 10) \), and again fill in the given values.
03

Calculate the percentage decrease

The percentage decrease in the intensity of the noise as a result of the installation of the muffler can be calculated using the formula \( percent\ decrease = \((I_{1} - I_{2}) / I_{1}\) * 100% \), and input the values of \( I_{1} \) and \( I_{2} \) obtained from step 1 and 2 respectively.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Decibels
Decibels are a measure for expressing the intensity of sound. They help us describe how loud or soft a sound is in a numerical value, making it easier to compare different sounds. One important thing to know about decibels ( hed standard abbreviation is dB) is that they do not represent a linear measure. This means that an increase by ten decibels does not double the noise.- The formula for finding the sound level \(\beta\), in decibels, is: \(\beta = 10 \log(I / I_{0})\).- \(I_{0}\) is the lowest intensity the human ear can hear, which is about \(10^{-12}\) watts per square meter.For example, if a sound is measured at 88 decibels before a change (like installing a muffler), it can be calculated into a physical sound intensity through this formula. By understanding decibels, students can connect numerical sound levels to real-world experiences of hearing.
Logarithmic Scale
A logarithmic scale is a type of scale used for a range of values that vary exponentially. In terms of sound intensity, the decibel scale is logarithmic. This allows for a wide range of sound levels to be expressed in a manageable way. - Logarithmic scales are beneficial because they can easily represent large differences in data. - With sound, every increase of 10 dB represents a tenfold increase in intensity. This characteristic is why two sounds that differ in intensity by just a few decibels can actually be quite different in terms of real-world loudness. Thanks to this scale, even intense sound waves can be communicated through relatively small numbers.
Percentage Decrease
When calculating a percentage decrease in sound intensity, you're essentially finding out how much the sound level has been reduced in comparison to its original level. This is particularly useful when measuring how effective a noise-reduction solution is, such as installing a muffler.- First, determine the initial and final sound intensities. In our case, before (88 dB) and after (72 dB) a muffler is installed.- Next, calculate their respective intensities using the formula \(I = I_{0} * 10^{(\beta / 10)}\) for both levels.- Finally, the percentage decrease can be computed with: \[(\text{Initial Intensity} - \text{Final Intensity}) / \text{Initial Intensity} \times 100\%\].This approach provides a clear picture of how much sound has been reduced, helping to understand the effectiveness of modifications made to reduce sound pollution.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 4 \\ \hline y & 5 & 1 \\ \hline \end{array} $$

The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution \(y=0.0266 e^{-(x-100)^{2} / 450}, 70 \leq x \leq 115,\) where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student.

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers \(y\) of cell sites from 1985 through 2008 can be modeled by \(y=\frac{237,101}{1+1950 e^{-0.355 t}}\) where \(t\) represents the year, with \(t=5\) corresponding to \(1985 .\) (Source: CTIA-The Wireless Association) (a) Use the model to find the numbers of cell sites in the years 1985,2000 , and 2006 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the number of cell sites will reach 235,000 . (d) Confirm your answer to part (c) algebraically.

The table shows the time \(t\) (in seconds) required for a car to attain a speed of \(s\) miles per hour from a standing start. $$ \begin{array}{|c|c|} \hline \text { Speed, } s & \text { Time, } t \\ \hline 30 & 3.4 \\ 40 & 5.0 \\ 50 & 7.0 \\ 60 & 9.3 \\ 70 & 12.0 \\ 80 & 15.8 \\ 90 & 20.0 \\ \hline \end{array} $$ Two models for these data are as follows. \(t_{1}=40.757+0.556 s-15.817 \ln s\) \(t_{2}=1.2259+0.0023 s^{2}\) (a) Use the regression feature of a graphing utility to find a linear model \(t_{3}\) and an exponential model \(t_{4}\) for the data. (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think best fits the data? Explain.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x-\ln (x+1)=2$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.