Problem 133 - Elementary and Intermediate Algebra

Question

In 1985 , Mexico City experienced an earthquake of magnitude 8.1 on the Richter scale. In \(1989,\) the San Francisco Bay area was rocked by an earthquake measuring \(7.1 .\) By what factor must the amplitude of an earthquake change to increase its severity by 1 point on the Richter scale? (Assume that the period remains constant.)

Step-by-Step Solution

Answer

The amplitude must increase by a factor of 10 for each 1-point increase on the Richter scale.

Step 1: Understanding the Richter Scale

The Richter scale measures the logarithmic scale of the amplitude of seismic waves. Each whole number increase on the Richter scale corresponds to a tenfold increase in amplitude of the seismic waves and approximately 31 times more energy release.

Step 2: Formula for Amplitude Factor

We know that the Richter scale is logarithmic. A difference of 1 on this scale corresponds to an amplitude that is 10 times greater. Therefore, if a magnitude increases by 1 point, the amplitude must increase by a factor of 10.

Step 3: Verification with Given Earthquake Magnitudes

To confirm this factor, consider the difference between the 1985 Mexico City earthquake (magnitude 8.1) and the 1989 San Francisco earthquake (magnitude 7.1). The difference in magnitude is 1 point (8.1 - 7.1 = 1.0). According to the formula, the amplitude of the Mexico City earthquake is 10 times that of the San Francisco earthquake if only considering the amplitude increase.

Key Concepts

Understanding Earthquake Amplitude

Earthquake amplitude is a measure of the strength or size of the seismic waves generated during an earthquake. Seismic waves are energy that travels through the Earth and is the direct cause of the ground shaking during an earthquake. The amplitude refers to the height of these waves when recorded by a seismograph, a device that detects and records seismic activity.
The size of an earthquake, as represented by its amplitude, directly affects how much ground shaking occurs. Larger amplitudes mean more intense shaking, which can lead to greater damage in affected areas. Amplitude is a key component in the calculation of an earthquake's magnitude using logarithmic scales like the Richter scale.
· Seismic waves cause the amplitude
· Larger amplitudes mean more intense shaking
· Amplitude is key to determining earthquake magnitude via logarithmic scales

The Logarithmic Scale and the Richter Scale

The Richter scale is a logarithmic scale used to quantify the magnitude of an earthquake. Unlike linear scales, where each step represents a uniform change, logarithmic scales increase exponentially. In the Richter scale, each whole number increase indicates a tenfold rise in seismic wave amplitude, showing the dramatic spike in energy released.
This means that an earthquake measured at magnitude 5 has waves 10 times greater in amplitude than one measured at magnitude 4. The energy release is even more significant, approximately 31 times more energy with each additional point increase. This exponential growth in amplitude and energy helps to reflect the true destructive potential of earthquakes.
· Logarithmic scales increase exponentially
· A 1-point increase on the Richter scale equals a tenfold increase in amplitude
· Energy release increases approximately 31 times per magnitude point increase

Seismic Waves and Their Impact

Seismic waves are the vibrations generated from the sudden breaking of rocks within the Earth or an explosion. They are crucial in understanding earthquakes because they radiate outward from the earthquake's focus, affecting areas far from the initial epicenter. These waves are what we feel during an earthquake; they're responsible for the shaking, rolling, and rattling observed during such events.
Seismic waves come in three main types: primary waves (P-waves), secondary waves (S-waves), and surface waves. P-waves are the fastest, traveling through both solids and liquids, followed by S-waves that move slower and only travel through solids. Surface waves travel the slowest but cause the most damage due to their rolling motion.
· Seismic waves are vibrations from rock breaks or explosions
· P-waves are fastest, followed by S-waves and then surface waves
· Surface waves cause most damage due to their rolling motion