Problem 66 - Precalculus: Functions and Graphs

Question

If an earthquake has a total horizontal displacement of \(S\) meters along its fault line, then the horizontal movement \(M\) of a point on the surface of Earth \(d\) kilometers from the fault line can be estimated using the formula $$M=\frac{S}{2}\left(1-\frac{2}{\pi} \tan ^{-1} \frac{d}{D}\right)$$ where \(D\) is the depth (in kilometers) below the surface of the focal point of the earthquake. Approximate the depth \(D\) of the focal point of an earthquake with \(S=3 \mathrm{m}\) if a point on the surface of Earth 5 kilometers from the fault line moved 0.6 meter horizontally.

Step-by-Step Solution

Answer

The focal point depth, \(D\), is approximately 2.16 kilometers.

Step 1: Identify Known Values

We are given:- Total displacement, \(S = 3\) meters.- Displacement at the surface, \(M = 0.6\) meters.- Distance from the fault line, \(d = 5\) kilometers.

Step 2: Set Up the Equation for Depth

Using the given formula:\[M = \frac{S}{2}\left(1-\frac{2}{\pi} \tan^{-1}\frac{d}{D}\right)\]Substitute \(M = 0.6\), \(S = 3\), and \(d = 5\) into the equation:\[0.6 = \frac{3}{2}\left(1-\frac{2}{\pi} \tan^{-1}\frac{5}{D}\right)\].

Step 3: Solve for the Inverse Tangent Expression

Firstly, simplify the equation:\[0.6 = 1.5\left(1-\frac{2}{\pi} \tan^{-1}\frac{5}{D}\right)\]Divide both sides by 1.5:\[\frac{0.6}{1.5} = 1 - \frac{2}{\pi} \tan^{-1}\frac{5}{D}\]Which simplifies to:\[0.4 = 1 - \frac{2}{\pi} \tan^{-1}\frac{5}{D}\].

Step 4: Isolate the Inverse Tangent Term

Rearrange the equation to isolate the inverse tangent term:\[\frac{2}{\pi} \tan^{-1}\frac{5}{D} = 1 - 0.4\]\[\frac{2}{\pi} \tan^{-1}\frac{5}{D} = 0.6\].

Step 5: Solve for Depth D Using Trigonometric Inversion

Multiply both sides by \(\pi/2\) to solve for the inverse tangent:\[\tan^{-1}\frac{5}{D} = \frac{\pi}{2} \times 0.6\]\[\tan^{-1}\frac{5}{D} = 0.3\pi\].

Step 6: Calculate Depth D

Use the tangent function to solve for \(D\):\[\frac{5}{D} = \tan(0.3\pi)\]\[D = \frac{5}{\tan(0.3\pi)}\]Calculate the approximate value for \(D\). "\(\tan(0.3\pi)\)" can be found using a calculator, and finally, \(D \approx 2.16\) kilometers.

Key Concepts

Seismic Waves

Seismic waves are the energy released during an earthquake, traveling through Earth's layers. These waves result from the sudden fracture and movement along faults and cause ground shaking. There are two main types of seismic waves: primary (P-waves) and secondary (S-waves):

• P-waves (Primary Waves): These are the fastest seismic waves and travel through both solid and liquid layers of Earth. They compress and expand the ground like an accordion.
• S-waves (Secondary Waves): Slower than P-waves, they can only move through solid material. S-waves cause more significant ground shaking and typically result in more damage.

Understanding seismic wave movement is essential for estimating earthquake impacts. Energy from these waves also helps locate an earthquake's origin, known as the epicenter, and its depth, called the focal depth.

Horizontal Displacement

Horizontal displacement refers to the lateral movement along a fault line during an earthquake. Imagine pushing the two sides of a cracked cookie apart; this gives you an idea of horizontal displacement. This parameter is crucial when studying seismic impact and potential structural damage.
Key factors influencing horizontal displacement include:

• The earthquake's magnitude and energy released.
• The type of fault (strike-slip, normal dip-slip, or reverse dip-slip).
• Distance from the fault line: Closer distances generally experience higher displacement.
• Subsurface geological conditions, such as rock type and structure.

In the exercise, the formula calculates how much surface movement (M) occurs due to horizontal displacement and other factors, informing us about the earthquake's potential effects at a given distance from the fault.

Focal Depth Calculation

Focal depth is the distance between Earth's surface and the earthquake's focal point. This is crucial for understanding the earthquake's potential impact. Shallow earthquakes tend to cause more surface damage than deeper ones.
The formula provided in the problem connects focal depth (D) to horizontal displacement at a specified distance. By rearranging the formula, you can solve for D given known variables.
Steps to calculate focal depth involve:

• Substitute known values into the formula.
• Isolate the term containing D.
• Use trigonometric functions to solve for D.

This understanding aids in hazard assessment and in refining our methods for predicting earthquake impacts.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \(\tan^{-1}\) (also called arctan), are used to find an angle when given two sides of a right triangle where one is opposite and the other is adjacent to the angle. In our exercise, the function \(\tan^{-1}(\frac{d}{D})\) helps solve for the unknown depth D by isolating and manipulating expressions!

• Why use inverse trigonometry? It allows solving for variables when direct measurements aren't feasible.
• Manipulating expressions: Rearrange terms to isolate the inverse tangent expression, enabling you to solve equations more easily.

This process forms the backbone of many calculations in earthquake mathematics, aiding in determining unknown values that impact seismic analysis and engineering solutions. The proper use of these functions enables accuracy in such critical calculations.