The Great Earthquake registered 8.8 on the Richter scale.
​(a) What was the intensity of the​ earthquake?
​(b) How many joules of energy was​ released?


The Richter scale measures the intensity of an earthquake \( I \) relative to a reference earthquake intensity \( I_0 \), defined by the equation:

\[
M = \log_{10} \left( \frac{I}{I_0} \right)
\]

Where:
- \( M \) is the magnitude on the Richter scale (in this case, 8.8),
- \( I \) is the intensity of the earthquake,
- \( I_0 \) is the reference intensity, often taken as the intensity of the weakest detectable earthquake.

### (a) **Finding the intensity \( I \)**:
Rearranging the formula to solve for \( I \):

\[
I = I_0 \cdot 10^M
\]

Given \( M = 8.8 \):

\[
I = I_0 \cdot 10^{8.8}
\]

Since \( I_0 \) is a constant reference value, the intensity of the earthquake is \( 10^{8.8} \) times more powerful than the reference intensity.

\[
I = 10^{8.8} \approx 6.31 \times 10^8 \cdot I_0
\]

Thus, the earthquake's intensity is approximately \( 6.31 \times 10^8 \) times greater than the reference earthquake.

### (b) **Finding the energy released in joules**:
The energy \( E \) in joules released by an earthquake can be related to its magnitude \( M \) on the Richter scale by the equation:

\[
\log_{10} E = 4.8 + 1.5M
\]

Substitute \( M = 8.8 \):

\[
\log_{10} E = 4.8 + 1.5 \times 8.8
\]
\[
\log_{10} E = 4.8 + 13.2 = 18.0
\]

Now, convert this back from logarithmic form:

\[
E = 10^{18.0} \text{ joules} = 1 \times 10^{18} \text{ joules}
\]

### **Answers**:
- (a) The intensity of the earthquake was approximately \( 6.31 \times 10^8 \cdot I_0 \).
- (b) The energy released was \( 1 \times 10^{18} \) joules.

Would you like further details or explanations?

Here are some related questions to consider:
1. How does the Richter scale compare with other earthquake intensity scales like the moment magnitude scale?
2. What is the physical meaning of the reference intensity \( I_0 \)?
3. How do Richter scale values correlate with the damage caused by an earthquake?
4. What are some historical examples of earthquakes with similar magnitudes?
5. How is energy release related to the earthquake’s depth and fault line movement?

**Tip:** The energy released by an earthquake increases exponentially with the magnitude, meaning a small increase in magnitude corresponds to a massive increase in energy.

The Great Earthquake registered 8.8 on the Richter scale. (a) What was the intensity of the earthquake? (b) How many joules of energy was released?

Solve the problem of determining the earthquake intensity and energy released for a magnitude of 8.8 on the Richter scale. This problem covers logarithmic functions, exponential calculations, and earthquake energy formulas. Ideal for students in Grades 10-12.

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