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To solve this problem, we are given the function for loudness in decibels:

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

where:
- \( D(I) \) is the decibel level,
- \( I \) is the intensity of the sound,
- \( 10^{-12} \) is the intensity of the quietest sound a human can hear.

We need to find the intensity \( I \) of the school fire alarm if it registers as 100 decibels. So, \( D(I) = 100 \).

### Steps to Solve:

1. **Set up the equation with \( D(I) = 100 \):**

   \[
   100 = 10 \log \left(\frac{I}{10^{-12}}\right)
   \]

2. **Isolate the logarithmic term by dividing both sides by 10:**

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

3. **Rewrite the equation in exponential form to solve for \( I \):**

   Recall that if \( \log(x) = y \), then \( x = 10^y \).

   So,

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

4. **Solve for \( I \) by multiplying both sides by \( 10^{-12} \):**

   \[
   I = 10^{10} \cdot 10^{-12}
   \]

5. **Simplify the expression:**

   \[
   I = 10^{-2}
   \]

So, the intensity \( I \) of the sound of the school fire alarm is \( 10^{-2} \) watts per square meter (W/m²).

---

Would you like further details on any of these steps, or do you have any questions?

### Additional Questions:
1. How would the intensity change if the decibel level was 110 dB?
2. Can you calculate the decibel level if the intensity was \( 10^{-3} \) W/m²?
3. What would the formula look like if the reference intensity changed?
4. How does a logarithmic scale help in measuring sound intensity?
5. What is the relationship between decibels and perceived loudness by humans?

### Tip:
In logarithmic equations, converting the logarithmic form to exponential form is often a useful technique for solving for unknown values.

What is the intensity of the sound of the school fire alarm if it registers as 100 decibels?

Learn how to calculate the intensity of a sound with a decibel level of 100 dB using logarithmic functions. This problem covers key concepts in logarithmic scales, sound intensity, and the decibel formula. Suitable for high school students in Grades 9-12.

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