The decibel level of a sound is given by the formula  D=10\log ((I)/(I_(0)))

where D
is the decibel level, I
is the intensity of the sound in watts per square meter, and
I_(0)​ is the reference intensity, 10^(-12)
watts per square meter.

Suppose the decibel level of a sound is 90dB
. What is the intensity of this sound?

,

To solve for the intensity \( I \) of the sound given the decibel level \( D = 90 \, \text{dB} \), we can use the given formula for the decibel level:

\[
D = 10 \log \left( \frac{I}{I_0} \right)
\]

where:

- \( D = 90 \, \text{dB} \) (decibel level),
- \( I_0 = 10^{-12} \, \text{W/m}^2 \) (reference intensity).

### Step 1: Plug in the given values.
Substitute \( D = 90 \) and \( I_0 = 10^{-12} \) into the equation:

\[
90 = 10 \log \left( \frac{I}{10^{-12}} \right)
\]

### Step 2: Isolate the logarithmic term.
Divide both sides of the equation by 10 to simplify:

\[
9 = \log \left( \frac{I}{10^{-12}} \right)
\]

### Step 3: Solve for the ratio \( \frac{I}{10^{-12}} \).
Rewrite the equation in exponential form (since \( \log \) is the inverse of exponentiation with base 10):

\[
10^9 = \frac{I}{10^{-12}}
\]

### Step 4: Solve for \( I \).
Multiply both sides by \( 10^{-12} \) to solve for \( I \):

\[
I = 10^9 \times 10^{-12} = 10^{-3} \, \text{W/m}^2
\]

### Final Answer:
The intensity of the sound is \( I = 10^{-3} \, \text{W/m}^2 \).

Would you like any further details or clarifications?

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Here are 5 related questions to further explore:

1. How does the intensity of sound change with a 10 dB increase?
2. What would be the intensity of a sound at 100 dB?
3. How would you convert between dB levels and intensity for sounds in everyday life?
4. What role does the reference intensity \( I_0 = 10^{-12} \) play in this formula?
5. How does this formula apply to different sources of noise, such as a conversation or machinery?

**Tip:** Decibels use a logarithmic scale, meaning each 10 dB increase corresponds to a tenfold increase in sound intensity.

The decibel level of a sound is given by the formula D = 10 log(I/I_0), where D is the decibel level, I is the intensity of the sound in watts per square meter, and I_0 is the reference intensity, 10^(-12) watts per square meter. Suppose the decibel level of a sound is 90dB. What is the intensity of this sound?

Learn how to calculate the intensity of sound from a given decibel level using the formula D = 10 log(I/I_0). This example covers the steps to find the intensity of a 90dB sound, using logarithmic properties and sound intensity concepts. Suitable for high school students in Grades 9-12.

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