Solved on Dec 07, 2023

Find the decibel level of a noise that produces 2.72×1052.72 \times 10^{-5} watt/m2^2 of power, given the formula D=10log(S/S0)\mathrm{D}=10 \mathrm{log}\left(\mathrm{S} / \mathrm{S}_{0}\right) where S0\mathrm{S}_{0} is 101210^{-12} watt/m2^2. (Round to the nearest decibel.)

STEP 1

Assumptions
1. The power of the noise, designated as SS, is 2.72×1052.72 \times 10^{-5} watt/m2^2.
2. The reference power level, designated as S0S_0, is 101210^{-12} watt/m2^2.
3. The formula to calculate the decibel level is D=10log(SS0)D = 10 \log\left(\frac{S}{S_0}\right).
4. The result should be rounded to the nearest decibel.

STEP 2

We will use the given formula to calculate the decibel level of the noise.
D=10log(SS0)D = 10 \log\left(\frac{S}{S_0}\right)

STEP 3

Substitute the given values for SS and S0S_0 into the formula.
D=10log(2.72×1051012)D = 10 \log\left(\frac{2.72 \times 10^{-5}}{10^{-12}}\right)

STEP 4

Calculate the ratio SS0\frac{S}{S_0}.
SS0=2.72×1051012\frac{S}{S_0} = \frac{2.72 \times 10^{-5}}{10^{-12}}

STEP 5

Perform the division to simplify the ratio.
SS0=2.72×105+12\frac{S}{S_0} = 2.72 \times 10^{-5 + 12}

STEP 6

Add the exponents in the ratio.
SS0=2.72×107\frac{S}{S_0} = 2.72 \times 10^7

STEP 7

Now, calculate the logarithm of the ratio.
log(2.72×107)\log\left(2.72 \times 10^7\right)

STEP 8

Use the property of logarithms that log(a×b)=log(a)+log(b)\log(a \times b) = \log(a) + \log(b).
log(2.72×107)=log(2.72)+log(107)\log\left(2.72 \times 10^7\right) = \log(2.72) + \log\left(10^7\right)

STEP 9

Calculate the logarithm of 10710^7.
log(107)=7\log\left(10^7\right) = 7

STEP 10

Use a calculator to find log(2.72)\log(2.72).
log(2.72)0.4343\log(2.72) \approx 0.4343

STEP 11

Add the logarithm values.
log(2.72)+log(107)0.4343+7\log(2.72) + \log\left(10^7\right) \approx 0.4343 + 7

STEP 12

Calculate the sum of the logarithm values.
log(2.72×107)0.4343+7=7.4343\log\left(2.72 \times 10^7\right) \approx 0.4343 + 7 = 7.4343

STEP 13

Multiply the logarithm by 10 to find the decibel level.
D=10×7.4343D = 10 \times 7.4343

STEP 14

Calculate the decibel level.
D10×7.4343=74.343D \approx 10 \times 7.4343 = 74.343

STEP 15

Round the decibel level to the nearest whole number.
D74D \approx 74
The decibel level of the noise is approximately 74 decibels.

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