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Chapter 12: Problem 14

At the race track, one race car starts its engine with a resulting intensity level of \(98.0 \mathrm{dB}\) at point \(P .\) Then seven more cars start their engines. If the other seven cars each produce the same intensity level at point \(P\) as the first car, what is the new intensity level with all eight cars running?

Short Answer

Expert verified
Answer: The new intensity level at point P with all eight cars running is approximately 109.0 dB.

Step by step solution

01

Recall the formula for intensity level in decibels

The formula for intensity level in decibels is $$ \beta = 10 \log_{10}\left(\frac{I}{I_0}\right), $$ where \(\beta\) is the intensity level in decibels, \(I\) is the intensity of the sound, and \(I_0\) is the reference intensity, which is \(10^{-12}\,\text{W/m}^2\).
02

Determine the relationship between intensity and the number of sound sources

When there are multiple sound sources producing the same intensity at a given point, the total intensity is the sum of the individual intensities of each source. In this case, there are eight cars running, and each one produces the same intensity, so the total intensity is: $$ I_\mathrm{total} = 8I_\mathrm{car}. $$
03

Calculate the intensity of one car in terms of the reference intensity

We are given that one car produces an intensity level \(98.0\,\text{dB}\) at point P. We substitute this value into the intensity level formula and solve for \(I_\mathrm{car}\): $$ 98.0 = 10 \log_{10}\left(\frac{I_\mathrm{car}}{I_0}\right). $$ Dividing both sides by 10, we have: $$ 9.8 = \log_{10}\left(\frac{I_\mathrm{car}}{I_0}\right). $$ Now we can find the intensity of one car as a multiple of the reference intensity using the inverse logarithm: $$ I_\mathrm{car} = I_0 \times 10^{9.8}. $$
04

Calculate the total intensity of all eight cars

Now we can find the total intensity of all eight cars using the relationship between intensity and the number of cars from Step 2: $$ I_\mathrm{total} = 8I_\mathrm{car} = 8 (I_0 \times 10^{9.8}). $$
05

Calculate the new intensity level of all eight cars running

Finally, we substitute the total intensity into the intensity level formula and solve for the new intensity level, \(\beta_\mathrm{total}\): $$ \beta_\mathrm{total} = 10 \log_{10}\left(\frac{8 (I_0 \times 10^{9.8})}{I_0}\right). $$ Simplify and calculate the new intensity level: $$ \beta_\mathrm{total} = 10\left(\log_{10}(8) + \log_{10}(10^{9.8})\right) = 10\left(\log_{10}(8) + 9.8\right) \approx 109.0\,\text{dB}. $$ Therefore, the new intensity level at point P with all eight cars running is approximately \(109.0\,\text{dB}\).

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