Problem 34 - Algebra and Trigonometry

Question

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 about 2512 times more intense than the Japanese earthquake.

Step 1: Understanding the Richter Scale

The Richter Scale is a logarithmic scale used to measure the magnitude of earthquakes, where each whole number increase represents a tenfold increase in measured amplitude and approximately 31.6 times more energy release. This means an earthquake of magnitude 5 is 10 times greater in amplitude than one of magnitude 4, and about 31.6 times more energy is released.

Step 2: Calculate the Difference in Magnitude

First, calculate the difference in magnitudes between the two earthquakes: \(8.3 - 4.9 = 3.4\). This tells us how many times greater the San Francisco earthquake is in magnitude than the Japanese earthquake.

Step 3: Determine Intensity Ratio Using Logarithmic Scale

The formula to find the intensity ratio based on the Richter scale is \(10^{m_1 - m_2}\), where \(m_1\) and \(m_2\) are the magnitudes of the two earthquakes. Substitute the magnitudes into the formula: \(10^{8.3 - 4.9} = 10^{3.4}\).

Step 4: Calculating the Intensity Ratio

Calculate \(10^{3.4}\) using a calculator, which results in approximately 2511.89. This value represents how many times more intense the San Francisco earthquake was compared to the Japanese earthquake.

Key Concepts

logarithmic scale

The Richter Scale is a perfect example of a logarithmic scale, which is often used in scientific measurements. In a logarithmic scale, each unit of increase corresponds to a multiplicative increase in the value being measured. For the Richter Scale, a one-unit increase in magnitude means a tenfold increase in the amplitude of the seismic waves. This makes it a very effective tool for comparing earthquake sizes, as even small differences in magnitude can represent large differences in seismic activity. This scale simplifies the wide range of earthquake energies into a more manageable and intuitive format. When looking at this scale, remember that a jump from a 7 to an 8 magnitude doesn't mean the earthquake is just a bit stronger; it's actually much stronger, with about 31.6 times more energy released.

earthquake magnitude

Earthquake magnitude is a measure of the energy released at the source of the earthquake. It is a key figure derived from the seismic waves detected by instruments all over the world. The magnitude tells us how "large" an earthquake is. Using the Richter Scale, the magnitude is represented as a single number. Each point on this scale corresponds to a specific amount of energy release. For example, a magnitude 5 earthquake releases approximately 31.6 times more energy than a magnitude 4 earthquake. This consistent rate makes it easier for scientists and researchers to quantify and compare the size of different earthquakes. So, when you hear about a magnitude 8.3 earthquake, like the one in San Francisco in 1906, you can understand it was much more powerful than an earthquake with a magnitude of 4.9, as it released much more energy.

intensity ratio

The intensity ratio gives us a way to compare the size of two earthquakes using their magnitudes. This ratio is calculated using the formula \[10^{m_1 - m_2}\]where \(m_1\) and \(m_2\) are the magnitudes of the two earthquakes. In this scenario, it's used to determine how many times more intense one earthquake is compared to another. For example, if you have an earthquake with a magnitude of 8.3 and another with 4.9, you subtract the magnitudes to find the difference, which is 3.4. You then calculate \[10^{3.4}\]which is approximately 2511.89. This tells you that the higher magnitude earthquake is 2511.89 times more intense than the lower one. This concept is particularly useful in understanding the massive differences in power between earthquakes that may not seem very different by their numerical magnitude alone.