A sound power level reading of 127 dB was taken near a construction site
where chippers were being used. When all but one of the chippers stopped working,
the sound power level reading was 120 dB. Estimate the number of chippers in
operation when the reading of 127 was obtained. Assume that the sources
may be treated as ideal point sources located at the same point.
To estimate the number of chippers in operation when the reading of 127 dB was obtained, we can use the concept of sound power level and the inverse square law for sound propagation.
The sound power level (L) is given in decibels (dB) and can be calculated using the formula:
L = 10 * log10(P/P0)
Where P is the sound power of the source and P0 is the reference sound power level.
In this case, we have two scenarios:
1. When all but one of the chippers are working, the sound power level reading is 120 dB.
2. When all the chippers are working, the sound power level reading is 127 dB.
Let's assume that each chipper produces the same sound power. So we can denote the sound power of one chipper as P.
Using the formula for sound power level, we can set up the following equations:
Equation 1: 120 = 10 * log10(n * P / P0)
Equation 2: 127 = 10 * log10((n + 1) * P / P0)
Where n is the number of chippers in operation.
Now, let's solve these equations to find the value of n:
Step 1: Divide both sides of Equation 1 by 10:
12 = log10(n * P / P0)
Step 2: Rewrite the equation in exponential form:
10^12 = n * P / P0
Step 3: Divide both sides of Equation 2 by 10:
12.7 = log10((n + 1) * P / P0)
Step 4: Rewrite the equation in exponential form:
10^12.7 = (n + 1) * P / P0
Step 5: Divide the equation obtained in Step 2 by the equation obtained in Step 4:
10^12 / 10^12.7 = (n * P / P0) / ((n + 1) * P / P0)
Step 6: Simplify the equation:
10^(-0.7) = n / (n + 1)
Step 7: Solve for n:
10^(-0.7) (n + 1) = n
10^(-0.7) n + 10^(-0.7) = n
10^(-0.7) = n - 10^(-0.7) n
10^(-0.7) = n (1 - 10^(-0.7))
n = 10^(-0.7) / (1 - 10^(-0.7))
Using a calculator, we can find the approximate value of n.
After calculating, we find that n is approximately 3.26.
Therefore, the estimate for the number of chippers in operation when the reading of 127 dB was obtained is 3.26. Since we cannot have a fractional number of chippers, we can round the estimate to the nearest whole number.
Therefore, the estimated number of chippers in operation when the reading of 127 dB was obtained is 3.