Chapter 3.5, Problem 63E

Solution

To identify: the correct answer.

The correct option is E $100$ .

Given information:

The 2001 earthquake in Arequipa, Peru $(R_1=8.1)$ .

The 1998 double earthquake in Takhar province, Afghanistan $(R_2=6.1)$ .

Explanation:

Let $a_1$ and $a_2$ represent the amplitude of the earthquakes in Peru and Afghanistan, respectively. Then compare the severity of the earthquakes by finding the ratio of their amplitudes, or namely:

  $\frac{a_1}{a_2}$

Write the equations for the two earthquakes:

  $8.1=\log \frac{a_1}{T}+B\cdots \cdots \cdots \cdots \cdots \cdots (1)$

  $6.1=\log \frac{a_2}{T}+B\cdots \cdots \cdots \cdots \cdots \cdots (2)$

Subtract equation $\left(2\right)$ from equation $\left(1\right)$

  $8.1-6.1=\log \frac{a_1}{T}-\log \frac{a_2}{T}$

Simplify and apply the Quotient Rule:

  $\begin{array}{c}2=\log \left[\frac{a_1}{T}\cdot \frac{T}{a_2}\right]\\ \log \frac{a_1}{a_2}=2\end{array}$

Rewrite in exponential form:

  $\begin{array}{c}\frac{a_1}{a_2}={10}^2\\ \frac{a_1}{a_2}=100\end{array}$

Therefore, the correct option is E $100$ .