Problem 36 - Algebra and Trigonometry

Question

The Northridge, California, earthquake of 1994 had a magnitude of 6.8 on the Richter scale. A year later, a 7.2-magnitude earthquake struck Kobe, Japan. How many times more intense was the Kobe earthquake than the Northridge earthquake?

Step-by-Step Solution

Answer

The Kobe earthquake was about 2.51 times more intense than the Northridge earthquake.

Step 1: Understanding the Richter Scale

The Richter scale is a logarithmic scale, meaning each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release. So, a 7.2 earthquake is significantly more intense than a 6.8.

Step 2: Calculate Magnitude Difference

Find the difference in magnitude between the Kobe and Northridge earthquakes: \(7.2 - 6.8 = 0.4\).

Step 3: Use the Logarithmic Exponentiation

Since the Richter scale is logarithmic, and each whole number represents a tenfold increase, to find how many times more intense, calculate \(10^{0.4}\). This represents the relative intensity between the two earthquakes.

Step 4: Calculate the Intensity Ratio

Compute \(10^{0.4}\). Using a calculator, you get approximately 2.51. This means the Kobe earthquake was approximately 2.51 times more intense than the Northridge earthquake.

Key Concepts

Logarithmic Scale

The Richter Scale is a fascinating method used to measure earthquake magnitudes. It utilizes a logarithmic scale. Instead of linear measurements that you might be familiar with, such as centimeters or inches, a logarithmic scale increases exponentially.

In this case, each number on the Richter scale represents a tenfold increase in the measured amplitude of an earthquake. This means that an earthquake with a magnitude of 7 is not just slightly stronger than a 6, but is significantly stronger.

The energy released by an earthquake, according to this scale, grows exponentially. That is why understanding the logarithmic nature of this scale is crucial when comparing different earthquake magnitudes.

Magnitude Difference

When it comes to comparing the strength of two earthquakes, the first step is to observe the difference in their magnitudes. In our example, the Kobe earthquake had a magnitude of 7.2, while the Northridge earthquake had a magnitude of 6.8.

To determine the magnitude difference, simply subtract the smaller magnitude from the larger one:

• Kobe: 7.2
• Northridge: 6.8
• Difference: 7.2 - 6.8 = 0.4

This simple calculation gives us the difference, which is 0.4. This number will be instrumental in the next steps to find out how many times more intense one earthquake is in comparison to the other.

Intensity Ratio

The intensity ratio reveals how much stronger one earthquake is compared to another. On the logarithmic Richter scale, each whole number increase equates to a tenfold increase in intensity. However, our difference of 0.4 isn’t a full magnitude increase.

To discover the intensity ratio, we calculate the exponentiation of 10 by the magnitude difference:

• Intensity Ratio Formula: \[ 10^{0.4} \]

Using a calculator, you will find that \( 10^{0.4} \approx 2.51 \).

This result indicates that the Kobe earthquake was approximately 2.51 times more intense than the Northridge earthquake.

Earthquake Intensity Calculation

Calculating the intensity of an earthquake involves understanding both the magnitude and the logarithmic nature of the Richter scale.

Start by determining the difference in magnitudes, which we found to be 0.4 between the Kobe and Northridge earthquakes. Then, compute the intensity ratio using the mathematical expression \( 10^{0.4} \).

This computation reflects how many times more intense one earthquake is than another. In this case, we've calculated that the Kobe quake was approximately 2.51 times more intense than the Northridge quake.

This method shows how crucial understanding logarithmic concepts is, as it allows us to comprehend and quantify the immense power of earthquakes through simple calculations.