Chapter 3.5, Problem 46E

Solution

To calculate : The number of times the 1995 Kobe, Japan earthquake was more severe than the 1994 Los Angeles earthquake.

The Kobe, Japan earthquake was about $4$ times greater as severe as the Los Angeles earthquake.

Given information :

  $R$ for Kobe, Japan earthquake was $7.2$ and $R$ for Los Angeles earthquake was $6.6$ .

Formula used :

  $R=\log \frac{a}{T}+B$ , where $a$ is the amplitude.

Calculation :

Let $a_1$ be the amplitude for the Kobe, Japan earthquake and $a_2$ be the amplitude for the Los Angeles earthquake.

Then,

  $\begin{array}{l}{R}_1=\log \frac{a_1}{T}+B=7.2\\ R_2=\log \frac{a_2}{T}+B=6.6\end{array}$

Find the ratio of severities $\frac{a_1}{a_2}$ :

  $\begin{array}{c}\left(\log \frac{a_1}{T}+B\right)-\left(\log \frac{a_2}{T}+B\right)=R_1-R_2\\ \log \frac{a_1}{T}-\log \frac{a_2}{T}=7.2-6.6\\ \log \frac{a_1}{a_2}=0.6\\ \frac{a_1}{a_2}={10}^{0.6}\\ a_1\approx 3.98107a_2\\ a_1\approx 4a_2\end{array}$

Hence, the Kobe, Japan earthquake was about $4$ times greater as severe as the Los Angeles earthquake.