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Chapter 16: Problem 28

At a state championship high school football game, the intensity level of the shout of a single person in the stands at the center of the field is about \(50 \mathrm{~dB}\). What would be the intensity level at the center of the field if all 10,000 fans at the game shouted from roughly the same distance away from that center point?

Short Answer

Expert verified
Answer: The combined intensity level at the center of the field when 10,000 fans shout simultaneously is 90 dB.

Step by step solution

01

Understand the Decibel Scale

The intensity level (IL) is measured in decibels (dB) and is calculated using the following formula: \(IL = 10 \times \log_{10} \frac{I}{I_0}\) Where \(I\) is the intensity of the sound and \(I_0\) is the reference intensity (\(10^{-12} W/m^2\)). To find the combined intensity of all the people shouting, we will need to first find the individual intensity, \(I_1\), of a single person shouting.
02

Find the Intensity of a Single Person’s Shout

Given the intensity level of a single person's shout (IL1) as 50 dB, we can use the formula to find the intensity: \(IL1 = 10 \times \log_{10} \frac{I_1}{I_0}\) Solve for \(I_1\): \(I_1 = I_0 \times 10^{\frac{IL1}{10}}\) Substitute the given values and calculate \(I_1\): \(I_1 = 10^{-12} W/m^2 \times 10^{\frac{50}{10}} = 10^{-12} W/m^2 \times 10^5 = 10^{-7} W/m^2\)
03

Calculate the Total Intensity of All Fans Shouting

Since there are 10,000 fans all shouting with the same intensity, we can simply add the individual intensity of each fan’s shout to find the total intensity at the center of the field: \(I_{total} = 10,000 \times I_1 = 10,000 \times 10^{-7} W/m^2 = 10^{-3} W/m^2\)
04

Find the Combined Intensity Level of all Fans

Now that we have the total intensity, we can use the formula for calculating the intensity level to find the combined intensity level (IL_total): \(IL_{total} = 10 \times \log_{10} \frac{I_{total}}{I_0}\) Substitute the values and calculate IL_total: \(IL_{total} = 10 \times \log_{10} \frac{10^{-3} W/m^2}{10^{-12} W/m^2} = 10 \times \log_{10} 10^{9} = 10 \times 9 = 90 \mathrm{~dB}\) So, the intensity level at the center of the field when all 10,000 fans shout from roughly the same distance away is 90 dB.

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Most popular questions from this chapter

Electromagnetic radiation (light) consists of waves. More than a century ago, scientists thought that light, like other waves, required a medium (called the ether) to support its transmission. Glass, having a typical mass density of \(\rho=2500 \mathrm{~kg} / \mathrm{m}^{3},\) also supports the transmission of light. What would the elastic modulus of glass have to be to support the transmission of light waves at a speed of \(v=2.0 \cdot 10^{8} \mathrm{~m} / \mathrm{s} ?\) Compare this to the actual elastic modulus of window glass, which is \(5 \cdot 10^{10} \mathrm{~N} / \mathrm{m}^{2}\).

A meteorite hits the surface of the ocean at a speed of \(8800 \mathrm{~m} / \mathrm{s}\). What are the shock wave angles it produces (a) in the air just before hitting the ocean surface, and (b) in the ocean just after entering? Assume the speed of sound in air and in water is \(343 \mathrm{~m} / \mathrm{s}\) and \(1560 \mathrm{~m} / \mathrm{s}\), respectively.

Compare the intensity of sound at the pain level, \(120 \mathrm{~dB}\), with that at the whisper level, \(20 \mathrm{~dB}\).

Two vehicles carrying speakers that produce a tone of frequency \(1000.0 \mathrm{~Hz}\) are moving directly toward each other. Vehicle \(\mathrm{A}\) is moving at \(10.00 \mathrm{~m} / \mathrm{s}\) and vehicle \(\mathrm{B}\) is moving at \(20.00 \mathrm{~m} / \mathrm{s}\). Assume the speed of sound in air is \(343.0 \mathrm{~m} / \mathrm{s}\), and find the frequencies that the driver of each vehicle hears.

The sound level in decibels is typically expressed as \(\beta=10 \log \left(I / I_{0}\right),\) but since sound is a pressure wave, the sound level can be expressed in terms of a pressure difference. Intensity depends on the amplitude squared, so the expression is \(\beta=20 \log \left(P / P_{0}\right),\) where \(P_{0}\) is the smallest pressure difference noticeable by the ear: \(P_{0}=2.00 \cdot 10^{-5} \mathrm{~Pa}\). A loud rock concert has a sound level of \(110 . \mathrm{dB}\), find the amplitude of the pressure wave generated by this concert.
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