Question

1. how much more powerful is a magnitude 6 earthquake than a magnitude 5 earthquake?
2. how does the intensity of a magnitude 8 earthquake compare to the intensity of a magnitude 5 earthquake?
3. by how much does the intensity of an earthquake increase as the magnitude of the earthquake increases by 1 on the richter scale?
4. the 1906 san francisco earthquake was one of the most destructive and infamous natural disasters in californias history. though the san francisco earthquake was powerful, the valdivia earthquake was 50 times more powerful. use this fact to determine the magnitude of the 1906 san francisco earthquake.

Explanation:

Step1: Recall Richter scale intensity rule

The Richter scale is logarithmic: each magnitude increase of 1 means intensity multiplies by $10^{\frac{3}{2}} = 10\sqrt{10} \approx 31.62$. For magnitude difference $n$, intensity ratio is $10^{\frac{3n}{2}}$.

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For Question 3:

Step1: Find magnitude difference

Magnitude difference $n = 6 - 5 = 1$

Step2: Calculate intensity ratio

$\text{Ratio} = 10^{\frac{3(1)}{2}} = 10\sqrt{10} \approx 31.62$

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For Question 4:

Step1: Find magnitude difference

Magnitude difference $n = 8 - 5 = 3$

Step2: Calculate intensity ratio

$\text{Ratio} = 10^{\frac{3(3)}{2}} = 10^{4.5} = 10^4 \times 10^{0.5} = 10000\sqrt{10} \approx 31622.78$

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For Question 5:

Step1: Use magnitude difference of 1

For $n=1$, intensity ratio is $10^{\frac{3(1)}{2}}$

Step2: Simplify the ratio

$10^{\frac{3}{2}} = 10\sqrt{10} \approx 31.62$

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For Question 6:

Note: The Valdivia earthquake has a known magnitude of 9.5. Let $M_S$ = San Francisco magnitude, $M_V=9.5$

Step1: Set up intensity ratio equation

$\text{Ratio} = 50 = 10^{\frac{3(M_V - M_S)}{2}}$

Step2: Take log base 10 of both sides

$\log_{10}(50) = \frac{3(9.5 - M_S)}{2}$

Step3: Solve for $M_S$

First, $\log_{10}(50) = \log_{10}(5 \times 10) = 1 + \log_{10}(5) \approx 1 + 0.6990 = 1.6990$
Rearrange: $9.5 - M_S = \frac{2 \times 1.6990}{3} \approx \frac{3.398}{3} \approx 1.1327$
$M_S = 9.5 - 1.1327 \approx 8.37$

Answer:

1. A magnitude 6 earthquake is approximately 31.62 times more powerful than a magnitude 5 earthquake.
2. A magnitude 8 earthquake has approximately 31622.78 times the intensity of a magnitude 5 earthquake.
3. The intensity increases by a factor of approximately 31.62 (or $10\sqrt{10}$) when magnitude increases by 1.
4. The magnitude of the 1906 San Francisco earthquake is approximately 8.4.