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Chapter 1: Problem 21

Fault rupture length: Earthquakes can result in various forms of damage, but many result in fault ruptures, or cracks in the Earth's surface. In a 1958 study 11 of earthquakes in California and Nevada, D. Tocher found the following relationship between the fault rupture length \(L\), in kilometers, and the magnitude \(M\) (on the Richter scale) of an earthquake: $$ L=0.0000017 \times 10.47^{M} . $$ What fault rupture length would be expected from an earthquake measuring \(6.5\) on the Richter scale?

Short Answer

Expert verified
The fault rupture length is approximately 6.119 km.

Step by step solution

01

Identify Known Values

We know the magnitude of the earthquake, which is given as \( M = 6.5 \). The given relationship between the fault rupture length \( L \) and magnitude \( M \) is \( L = 0.0000017 \times 10.47^{M} \). Our task is to find \( L \) given \( M = 6.5 \).
02

Substitute the Known Magnitude

Substitute the known magnitude \( M \) into the equation. The equation becomes:\[ L = 0.0000017 \times 10.47^{6.5} \]
03

Calculate the Exponential Expression

Calculate \( 10.47^{6.5} \). You can use a calculator for this part. After computation, \( 10.47^{6.5} \approx 3599309.59 \).
04

Calculate the Fault Rupture Length

Substitute the value obtained from the exponential calculation back into the equation:\[ L = 0.0000017 \times 3599309.59 \]Multiply the values together to find \( L \), which gives approximately \( L \approx 6.119 \) km.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Richter Scale
The Richter Scale is a logarithmic scale used to quantify the magnitude of an earthquake. Unlike linear scales, each whole number increase on the Richter Scale represents a tenfold increase in the measured seismic amplitude.
A small increase can mean significant growth in the energy release of an earthquake, affecting its potential damage. Invented in 1935 by Charles F. Richter, it's been instrumental in helping seismologists understand and compare different seismic events.
  • The Richter Scale helps determine earthquake strength and potential destruction.
  • It is essential for assessing areas at risk and planning construction specifications.
  • Despite newer scales, it remains commonly used for its straightforwardness.
The Richter Scale's practicality lies in its ability to compress various earthquake sizes into an understandable measure.
Fault Rupture Length
Fault rupture length indicates the linear extent of displacement along a fault line. When an earthquake occurs, it triggers a rupture, creating cracks or fractures along a fault.
Understanding the rupture length is vital for estimating the potential impact of an earthquake.
The relationship between fault rupture length and earthquake magnitude helps predict the physical surface changes caused by seismic activities.
  • Longer rupture lengths tend to correlate with stronger and potentially more destructive earthquakes.
  • Fault rupture length can influence rebuilding efforts after an earthquake.
  • Monitoring such lengths helps design robust infrastructure, resilient to seismic activities.
In seismology, studying rupture length is key to understanding the mechanics of earthquakes and their impacts.
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of earthquakes, these functions are often used to model relationships, like the one between earthquake magnitude and fault rupture length.
For instance, in our exercise, the equation used is:\[ L = 0.0000017 \times 10.47^{M} \]where the variable exponent (earthquake magnitude) significantly influences the resulting value.
Key aspects of exponential functions include:
  • Rapid growth or decay, depending on the base and exponent.
  • Critical in modeling real-world phenomena, especially where change is non-linear.
  • Often used in engineering and natural sciences to predict outcomes and analyze trends.
Understanding exponential functions is crucial for interpreting various natural and engineered systems.
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth. It covers the monitoring, measurement, and analysis of seismic activity to better understand the Earth's behavior.
As a branch of earth sciences, it combines physics and geology, using tools like seismographs and mathematical models for earthquake analysis.
  • Seismologists aim to predict earthquakes, minimizing human and economic risks.
  • They study the Earth's layers and how seismic waves travel through them.
  • Seismology plays a crucial role in understanding plate tectonics and the Earth's structure.
Through such detailed studies, seismology helps anticipate seismic events, guiding the development of safety measures and urban planning strategies to mitigate potential catastrophes.

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Most popular questions from this chapter

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