Solved on Dec 07, 2023

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

STEP 1

Assumptions
1. The power of the noise, designated as $S$ , is $2.72 \times 10^{-5}$ watt/m $^2$ .
2. The reference power level, designated as $S_0$ , is $10^{-12}$ watt/m $^2$ .
3. The formula to calculate the decibel level is $D = 10 \log\left(\frac{S}{S_0}\right)$ .
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.
$D = 10 \log\left(\frac{S}{S_0}\right)$

STEP 3

Substitute the given values for $S$ and $S_0$ into the formula.
$D = 10 \log\left(\frac{2.72 \times 10^{-5}}{10^{-12}}\right)$

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

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

STEP 14

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

STEP 15

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

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

STEP 1

STEP 2

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

STEP 3

Substitute the given values for $S$ and $S_0$ into the formula.
$D = 10 \log\left(\frac{2.72 \times 10^{-5}}{10^{-12}}\right)$

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

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

STEP 14

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

STEP 15

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

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STEP 1

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)D=10log(S0​S​)

STEP 3

Substitute the given values for SSS and S0S_0S0​ into the formula.D=10log⁡(2.72×10−510−12)D = 10 \log\left(\frac{2.72 \times 10^{-5}}{10^{-12}}\right)D=10log(10−122.72×10−5​)

STEP 4

Calculate the ratio SS0\frac{S}{S_0}S0​S​.SS0=2.72×10−510−12\frac{S}{S_0} = \frac{2.72 \times 10^{-5}}{10^{-12}}S0​S​=10−122.72×10−5​

STEP 5

Perform the division to simplify the ratio.SS0=2.72×10−5+12\frac{S}{S_0} = 2.72 \times 10^{-5 + 12}S0​S​=2.72×10−5+12

STEP 6

Add the exponents in the ratio.SS0=2.72×107\frac{S}{S_0} = 2.72 \times 10^7S0​S​=2.72×107

STEP 7

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

STEP 8

Use the property of logarithms that log⁡(a×b)=log⁡(a)+log⁡(b)\log(a \times b) = \log(a) + \log(b)log(a×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)log(2.72×107)=log(2.72)+log(107)

STEP 9

Calculate the logarithm of 10710^7107.log⁡(107)=7\log\left(10^7\right) = 7log(107)=7

STEP 10

Use a calculator to find log⁡(2.72)\log(2.72)log(2.72).log⁡(2.72)≈0.4343\log(2.72) \approx 0.4343log(2.72)≈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 + 7log(2.72)+log(107)≈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 = 7.4343log(2.72×107)≈0.4343+7=7.4343

STEP 13

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

STEP 14

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

STEP 15

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

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

Substitute the given values for $S$ and $S_0$ into the formula.
$D = 10 \log\left(\frac{2.72 \times 10^{-5}}{10^{-12}}\right)$

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

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

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

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

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

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

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

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

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

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

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

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