Problem 98 - Applied Calculus

Question

GENERAL: Earthquakes The sizes of major earthquakes are measured on the Moment Magnitude Scale, or MMS, although the media often still refer to the outdated Richter scale. The MMS measures the total energy relensed by an earthquake, in units denoted \(M_{W}\) (W for the work accomplished). An increase of \(1 M_{W}\) means the energy increased by a factor of \(32,\) so an increase from \(A\) to \(B\) means the energy increased by a factor of \(32^{B-A}\). Use this formula to find the increase in energy between the following earthquakes: The 2001 earthquake in India that measured \(7.7 M_{W}\) and the 2011 earthquake in Japan that measured \(9.0 M_{W}\). (The earthquake in Japan generated a 28 -foot tsunami wave that traveled six miles inland, killing 24,000 and causing an estimated \(\$ 300\) billion in damage, making it the most expensive natural disaster ever recorded.)

Step-by-Step Solution

Answer

The energy increased by approximately 56.23 times.

Step 1: Identify Given Values

We are given two earthquakes: one in 2001 with an intensity of \(7.7 \, M_{W}\), and one in 2011 with an intensity of \(9.0 \, M_{W}\). We need to calculate the energy increase factor between these two magnitudes.

Step 2: Define the Increase Formula

The formula to calculate the increase in energy is \(32^{(B - A)}\), where \(A\) and \(B\) are the moment magnitudes of the two earthquakes. For our exercise, \(A = 7.7\) and \(B = 9.0\).

Step 3: Calculate the Difference in Magnitude

Subtract the smaller magnitude from the larger one: \(B - A = 9.0 - 7.7 = 1.3\).

Step 4: Substitute Values into the Formula

Using the difference calculated in the previous step, substitute into the energy increase formula: \(32^{1.3}\).

Step 5: Calculate the Energy Increase Factor

Compute \(32^{1.3}\) using a calculator: \(32^{1.3} \approx 56.23\).

Step 6: Interpret the Result

The energy release of the 2011 Japan earthquake was approximately 56.23 times greater than the 2001 India earthquake.

Key Concepts

Energy Increase Factor

When we talk about earthquakes, understanding how much energy is released is crucial. The Moment Magnitude Scale, or MMS, is a tool to measure this energy. Here, the term **energy increase factor** comes into play. It tells us how much more energy one earthquake releases compared to another.

Imagine each magnitude point on the MMS being like a step up on a ladder. But, this is no ordinary ladder. Each step represents an increase that's much larger than the last. When the magnitude increases by 1 unit, the energy released isn't just a little bit more; it's 32 times more!

To find out how much more energy one earthquake had over another, we use a formula:

• Calculate the difference in magnitudes between the two earthquakes.
• Use this difference to find the energy increase: \(32^{(B - A)}\), where \(B\) is the larger magnitude and \(A\) the smaller magnitude.

This factor helps us understand the immense power of nature better.

Earthquake Measurement

Earthquakes can be terrifying, but how do we compare their magnitudes? The Moment Magnitude Scale (MMS) is the go-to measure used by scientists to gauge an earthquake's potency. Unlike older scales, MMS calculates the total energy released, giving us a more comprehensive understanding of an earthquake's strength.

The scale's unit of measure is \( M_{W} \), showcasing the idea of "work" or "energy." It's a vast improvement over the outdated Richter scale, which mainly focused on amplitude of waves instead of total energy.

In reality, magnitude numbers can seem abstract. But with MMS, we understand not just whether an earthquake is big or small, but also how it compares to other quakes in terms of energy released, helping predict possible impacts. Students learning about MMS should focus on its ability to convey more detailed information than other scales used before.

Logarithmic Scale

At first glance, the logarithmic scale used in measuring earthquakes might seem complicated. But breaking it down reveals its importance. This scale isn't linear; it's exponential.

Here's a simple way to think about it:

• If you increase the magnitude by just 1, it's like multiplying the energy by a certain consistent factor (32, in the case of MMS).
• This means an earthquake that measures 6.0 on the scale is not twice as powerful as one measuring 3.0; it's exponentially greater!

By using a logarithmic scale, scientists manage to bring very large numbers into a more manageable range.

Ultimately, this scale makes it easier to compare the differences in energy released by various earthquakes—emphasizing how dramatic even small changes in magnitude can be.