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Select the correct answer from the drop-down menu.The formula 𝛽=10⁢log⁡(𝐼𝐼0) is used to find the sound level, 𝛽 , in decibels (dB), of a sound with an intensity of I. In the formula, 𝐼0 represents the smallest sound intensity that can be heard by the human ear (approximately 10-12watts/meter2 ).The sound level on a busy street is 70 dB. What is the sound intensity on the street?The sound intensity on the busy street is watts/meter2 .

Question

Select the correct answer from the drop-down menu.The formula 𝛽=10⁢log⁡(𝐼𝐼0) is used to find the sound level, 𝛽 , in decibels (dB), of a sound with an intensity of I. In the formula, 𝐼0 represents the smallest sound intensity that can be heard by the human ear (approximately 10-12watts/meter2 ).The sound level on a busy street is 70 dB. What is the sound intensity on the street?The sound intensity on the busy street is watts/meter2 .

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Solution

To find the sound intensity I I on a busy street given the sound level β \beta of 70 dB, we can use the formula:

β=10log⁡(II0) \beta = 10 \log \left( \frac{I}{I_0} \right)

where:

  • β \beta is the sound level in decibels (dB),
  • I I is the sound intensity,
  • I0 I_0 is the reference sound intensity, which is 10−12 10^{-12} watts/meter2^2.

Given: β=70 dB \beta = 70 \text{ dB} I0=10−12 watts/meter2 I_0 = 10^{-12} \text{ watts/meter}^2

We need to solve for I I . Follow these steps:

  1. Substitute the given values into the formula: 70=10log⁡(I10−12) 70 = 10 \log \left( \frac{I}{10^{-12}} \right)

  2. Divide both sides of the equation by 10 to isolate the logarithm: 7=log⁡(I10−12) 7 = \log \left( \frac{I}{10^{-12}} \right)

  3. Rewrite the equation in exponential form to solve for I I : 107=I10−12 10^7 = \frac{I}{10^{-12}}

  4. Multiply both sides by 10−12 10^{-12} to isolate I I : I=107×10−12 I = 10^7 \times 10^{-12}

  5. Simplify the expression: I=107−12 I = 10^{7 - 12} I=10−5 I = 10^{-5}

Therefore, the sound intensity on the busy street is 10−5 10^{-5} watts/meter2^2.

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