Question

A formula for calculating the magnitude of an earthquake is M=2/3*log(E/E[0]) that uses the common logarithm. This is called moment magnitude scale, an alternative to the more well known Richter scale. One earthquake has a magnitude of 3.9 on the MMS. If a second earthquake has 800 times as much energy as the first, find the magnitude of the second quake

Answer 1

To find the magnitude of the second earthquake, substitute the energy of the second earthquake into the formula M=2/3*log(E/E[0]).

Step-by-step explanation:

To find the magnitude of the second earthquake, we can use the formula M=2/3*log(E/E[0]), where M is the magnitude, E is the energy, and E[0] is a reference energy. The second earthquake has 800 times as much energy as the first earthquake, so we can substitute 800E[0] for E. Plugging this into the formula, we get M = 2/3 * log(800E[0]/E[0]) = 2/3 * log(800) = 2/3 * 2.9031 ≈ 1.935. Therefore, the magnitude of the second earthquake is approximately 1.935.

Answer 2

The magnitude of the second earthquake, is approximately 5.84.

To find the magnitude of the second earthquake, we'll use the given moment magnitude scale (MMS) formula and the information about the earthquakes' energies. The MMS formula is:

`\[ M = (2)/(3) \log\left((E)/(E_0)\right) \]`

where:

- M is the magnitude,

- E is the energy released by the earthquake,

- E0 is a reference energy level,

- log represents the common logarithm (base 10).

Given that the first earthquake has a magnitude of 3.9 and the second earthquake has 800 times as much energy as the first, we can find the magnitude of the second earthquake as follows:

Step 1: Express the Magnitude of the First Earthquake

For the first earthquake with magnitude 3.9:

`\[ 3.9 = (2)/(3) \log\left((E_1)/(E_0)\right) \]`

where
`\( E_1 \)` is the energy of the first earthquake.

Step 2: Solve for
`\( (E_1)/(E_0) \)`

Rearranging the formula, we get:

`\[ \log\left((E_1)/(E_0)\right) = (3)/(2) * 3.9 \]`

We can then find
`\( (E_1)/(E_0) \)` by taking the antilogarithm.

Step 3: Calculate the Energy Ratio for the Second Earthquake

The energy of the second earthquake is 800 times that of the first earthquake, so
`\( E_2 = 800 * E_1 \).` The magnitude of the second earthquake,
`\( M_2 \)`, is:

`\[ M_2 = (2)/(3) \log\left((E_2)/(E_0)\right) = (2)/(3) \log\left((800E_1)/(E_0)\right) \]`

Step 4: Express
`\( M_2 \) in Terms of \( M_1 \)`

We can use the relationship between
`\( E_1 \) and \( E_0 \)` from the first earthquake to express
`\( M_2 \) in terms of \( M_1 \) (3.9).`

Let's start by calculating
`\( (E_1)/(E_0) \)`and then use it to find the magnitude of the second earthquake.

The magnitude of the second earthquake, which has 800 times the energy of the first earthquake with a magnitude of 3.9 on the Moment Magnitude Scale (MMS), is approximately 5.84.