Problem 36 - Precalculus Mathematics for Calculus

Question

These exercises deal with logarithmic scales. The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time in Japan an earthquake with magnitude 4.9 caused only minor damage. How many times more intense was the San Francisco earthquake than the Japanese earthquake?

Step-by-Step Solution

Answer

The San Francisco earthquake was approximately 2511.886 times more intense than the Japanese earthquake.

Step 1: Understanding the Richter Scale

The Richter scale is a logarithmic scale used to measure the intensity of earthquakes. For each whole number increase on the Richter scale, the intensity of the earthquake is 10 times greater.

Step 2: Expression for Intensities

Let \( I_S \) be the intensity of the San Francisco earthquake with magnitude 8.3 and \( I_J \) be the intensity of the Japanese earthquake with magnitude 4.9. The intensity is defined as \( I = 10^{M} \), where \( M \) is the magnitude.

Step 3: Calculate San Francisco's Intensity

Using the equation \( I = 10^M \), the intensity of the San Francisco earthquake is \( I_S = 10^{8.3} \).

Step 4: Calculate Japan's Intensity

Similarly, the intensity of the Japanese earthquake is \( I_J = 10^{4.9} \).

Step 5: Determine the Intensity Ratio

To find how many times more intense the San Francisco earthquake was compared to the Japanese earthquake, calculate the ratio \( \frac{I_S}{I_J} = \frac{10^{8.3}}{10^{4.9}} \).

Step 6: Simplify the Intensity Ratio

Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), simplify the ratio: \( 10^{8.3 - 4.9} = 10^{3.4} \).

Step 7: Final Calculation

Calculate \( 10^{3.4} \) using a calculator to find the precise ratio of intensities. \( 10^{3.4} \approx 2511.886 \).

Key Concepts

Logarithmic Scales

Logarithmic scales are a fascinating concept in mathematics, and they are especially useful when dealing with data that spans multiple orders of magnitude, like earthquake intensities. Instead of increasing linearly, these scales increase in a log manner, meaning each step represents an exponential growth. This is why a Richter scale of 7 is not just one unit bigger than a 6; it's actually ten times more intense.

Let's break this down:

• On a linear scale, each step increases by a constant amount, like 1, 2, 3.
• A logarithmic scale increases by powers of a base number, often 10, such as 10, 100, 1000.
• This makes it easier to represent very large numbers without using a huge range of numbers on our graph.

In practical terms, if an earthquake jumps from a magnitude of 5 to 6, the ground moves ten times as much and releases about 31.6 times more energy. Logarithmic scales help manage these high-intensity measurements in a straightforward way.

Earthquake Intensity

Earthquake intensity is a measure of the effects of an earthquake at different locations over its range. Unlike the magnitude, which stays constant no matter where you measure it, the intensity can vary depending on your distance from the epicenter.To quantify earthquake intensity using the Richter scale:

• Magnitude is determined by the logarithm of the amplitude of waves measured by seismographs.
• The formula for intensity using magnitude (M) is given by: \[ I = 10^M \]

This means the intensity is driven by a power of 10. So a small change in magnitude results in a large change in intensity. For example, the magnitude 8.3 earthquake in San Francisco meant immense energy release compared to the magnitude 4.9 quake in Japan, resulting in very different impacts in their respective regions.

Intensity Ratio

The intensity ratio allows us to directly compare two earthquakes and understand the difference in their power. Using the properties of exponents can make these calculations straightforward, even though the numbers involved are huge.Here’s how it works:

• Determine the intensity for each event using the formula \( I = 10^M \).
• The ratio is \( \frac{I_{S}}{I_{J}} \), where \( I_{S} \) and \( I_{J} \) are the intensities of San Francisco and Japan earthquakes respectively.
• This simplifies to \( 10^{M_S - M_J} \)= \( 10^{8.3 - 4.9} \).
• This step involves subtracting the magnitudes: 8.3 - 4.9 = 3.4, thus expediting the calculation.
• Finally, compute \( 10^{3.4} \) using a calculator for precision, resulting in approximately 2511.89.

This means that the San Francisco earthquake was about 2,512 times more intense than the one in Japan, demonstrating the dramatic effects of even a few points' difference on the Richter scale.