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The problem provides the logarithmic formula for calculating sound intensity in decibels:

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

where:
- \( I \) is the sound intensity in watts per square meter,
- \( I_0 = 10^{-12} \) watts per square meter (the reference intensity for the lowest audible sound).

You are asked to find the decibel level \( S(x) \) for an airplane with a sound intensity of \( I = 8.3 \times 10^2 \) watts per square meter.

### Step-by-step Solution:

1. Plug the given values into the formula:
   \[
   S(x) = 10 \log\left(\frac{8.3 \times 10^2}{10^{-12}}\right)
   \]

2. Simplify the argument of the logarithm:
   \[
   \frac{8.3 \times 10^2}{10^{-12}} = 8.3 \times 10^{14}
   \]

3. Now calculate the logarithm:
   \[
   S(x) = 10 \log(8.3 \times 10^{14})
   \]

   Using the property of logarithms \( \log(a \times b) = \log(a) + \log(b) \):
   \[
   \log(8.3 \times 10^{14}) = \log(8.3) + \log(10^{14})
   \]
   \[
   \log(8.3) \approx 0.919
   \]
   \[
   \log(10^{14}) = 14
   \]
   So,
   \[
   \log(8.3 \times 10^{14}) = 0.919 + 14 = 14.919
   \]

4. Multiply by 10:
   \[
   S(x) = 10 \times 14.919 = 149.19
   \]

Thus, the sound intensity emitted by the airplane is approximately **149 decibels**.

Would you like any further details or explanations?

Here are 5 related questions to expand your understanding:
1. How does changing the reference intensity \( I_0 \) affect the decibel calculation?
2. What is the physical meaning of a decibel value?
3. How is the logarithmic scale useful for measuring sound intensity?
4. What are typical decibel levels for various everyday sounds?
5. How would you calculate the decibel level if the intensity was given in different units?

**Tip**: The decibel scale is logarithmic, meaning a small increase in decibel level represents a large increase in intensity.

How many decibels are emitted from an airplane with a sound intensity of 8.3 * 10^2 watts per square meter, using the logarithmic model for sound intensity S(x) = 10 log(I/I_0) where I_0 = 10^(-12)?

Find out how to calculate the sound intensity in decibels from an airplane emitting 8.3 * 10^2 watts per square meter using the logarithmic decibel formula S(x) = 10 log(I/I_0). This problem is ideal for students in Grades 11-12.

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