Studdy AI Logo
Math

Question Find the decibel level of a noise that produces 2.72×10−52.72 \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 10−1210^{-12} watt/m2^2. (Round to the nearest decibel.)

Studdy Solution

STEP 1

Assumptions
1. The power of the noise, designated as SS, is 2.72×10−52.72 \times 10^{-5} watt/m2^2.
2. The reference power level, designated as S0S_0, is 10−1210^{-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⁡(2.72×10−510−12)D = 10 \log\left(\frac{2.72 \times 10^{-5}}{10^{-12}}\right)

STEP 4

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

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}

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.
D≈10×7.4343=74.343D \approx 10 \times 7.4343 = 74.343

STEP 15

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

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord