Chapter 8.4, Problem 34PPS

a.

To determine

The scale rating number of times as great is the amplitude.

Answer to Problem 34PPS

  $1,000$ Times.

Explanation of Solution

Given information:

The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude M is given by $M={\log }_{10}x$ , where $x$ represents the amplitude of the seismic wave causing ground motion.

  $(a)$ How many times as great is the amplitude caused by an earthquake with a Richter scale rating of $8$ as an aftershock with a Richter scale rating of $5?$

Calculation:

From the given information, the amplitude of the seismic wave with a Richter scale rating of $8$ and $5$ are ${10}^8$ and ${10}^5$ respectively. Divide ${10}^8$ by ${10}^5$ :

  $\frac{{10}^8}{{10}^5}={10}^{8-5}={10}^3=1,000$

Therefore, the scale rating of $8$ is $1,000$ times greater than the scale rating of $5$ .

b.

To determine

The number of times greater the New Madrid earthquake than the San Francisco earthquake.

Answer to Problem 34PPS

The New Madrid earthquake was ${10}^{0.3}$ times greater than the San Francisco earthquake.

Explanation of Solution

Given information:

The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude M is given by $M={\log }_{10}x$ , where $x$ represents the amplitude of the seismic wave causing ground motion.

  $(b)$ In $1906,$ San Francisco was almost completely destroyed by a $7.8$ magnitude earthquake. In $1911,$ an earthquake estimated at magnitude $8.1$ occurred along the New Madrid fault in the Mississippi River Valley. How many times greater was the New Madrid earthquake than the San Francisco earthquake?

Calculation:

The amplitude of San Francisco earthquake was ${10}^{7.8}$ and the amplitude of New Madrid earthquake was ${10}^{8.1}$ . Divide ${10}^{8.1}$ by ${10}^{7.8}$ :

  $\begin{array}{l}\frac{{10}^{8.1}}{{10}^{7.8}}={10}^{8.1-7.8}\\ ={10}^{0.3}\end{array}$

The New Madrid earthquake was ${10}^{0.3}$ times greater than the San Francisco earthquake.