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To solve part 1b, we are given the equation that relates the energy released by a quake \( y \) to its magnitude \( x \) on the Richter scale:

\[
\frac{2}{3} \log_{10} y = x
\]

Where:
- \( y \) is the energy released,
- \( x \) is the magnitude on the Richter scale.

We need to determine the relative energy released by the Chilean earthquake (magnitude 8.8) compared to the Haitian earthquake (magnitude 7.0). This involves calculating the ratio of the energy released by the two quakes.

### Step 1: Rearranging the formula to solve for \( y \)
We can solve for \( y \) by isolating it:

\[
\log_{10} y = \frac{3}{2}x
\]
\[
y = 10^{\frac{3}{2}x}
\]

### Step 2: Calculate energy for both earthquakes

For the Chilean quake (\( x = 8.8 \)):
\[
y_{\text{Chile}} = 10^{\frac{3}{2} \times 8.8} = 10^{13.2}
\]

For the Haitian quake (\( x = 7.0 \)):
\[
y_{\text{Haiti}} = 10^{\frac{3}{2} \times 7.0} = 10^{10.5}
\]

### Step 3: Calculate the ratio of energies

Now, we calculate how many times stronger the Chilean quake was compared to the Haitian quake by dividing the energy of the Chilean quake by the energy of the Haitian quake:

\[
\frac{y_{\text{Chile}}}{y_{\text{Haiti}}} = \frac{10^{13.2}}{10^{10.5}} = 10^{13.2 - 10.5} = 10^{2.7}
\]

### Step 4: Final result

Thus, the Chilean earthquake released approximately \( 10^{2.7} \) times more energy than the Haitian earthquake. Calculating \( 10^{2.7} \):

\[
10^{2.7} \approx 501.19
\]

So, the Chilean earthquake released about **501 times** more energy than the Haitian earthquake.

Would you like more details on the steps, or have any questions?

### 5 Related Questions:
1. How does the Richter scale's logarithmic nature affect the interpretation of earthquake magnitudes?
2. What is the significance of using base-10 logarithms in seismic measurements?
3. Can the energy released by an earthquake be calculated if its magnitude is known?
4. How does the energy formula for earthquakes compare to the magnitude formula?
5. What are the implications of energy release on earthquake damage in populated areas?

### Tip:
When working with logarithmic scales, small changes in magnitude correspond to large differences in the actual quantities, as the scale grows exponentially.

Another measure of a quake’s strength is related to the energy released by a quake. One of these measures is given by the equation (2/3) log₁₀ y = x, where x is the number reported for a quake and y is the energy released by the quake. What is the relative magnitude of the Chilean and Haitian quake (based on energy released)? That is, how many times stronger (not measured in logs but in terms of the underlying energy released)?

Solve the relative energy difference between the Chilean and Haitian earthquakes based on seismic energy formulas. Learn how to apply logarithmic equations to compare two earthquakes, using the formula (2/3) log₁₀ y = x and exponential growth. Suitable for Grades 10-12.

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