Chapter 2.2, Problem 96E

Earthquake Intensity The magnitude of an earthquake, measured on the Richter scale, is given by

$R\left(I\right)=\log \frac{I}{I_0}$,

where I is the amplitude registered on a seismograph located $100$ Km from the epicenter of the earthquake, and $I_0$ is the amplitude of a certain small size earthquake. Find the Richter scale ratings of earthquakes with the following amplitudes.

a. $\mathrm{1,000,000}{I}_0$ b. $\mathrm{100,000,000}{I}_0$

c. On June $15, 1999$, the city of Puebla in central Mexico was shaken by an earthquake that measured $6.7$ on the Richter scale. Express this reading in terms of $I_0$. Source: Exploring Colonial Mexico.

d. On September $19, 1985$, Mexico's largest recent earthquake, measuring $8.1$ on the Richter scale, killed about $\mathrm{10,000}$ people. Express the magnitude of an $8.1$ reading in terms of $I_0.$Source: History.com.

e. Compare your answers to part c and d. How much greater was the force of the $1985$ earthquake than the $1999$ earthquake?

f. The relationship between the energy E of an earthquake and the magnitude on the Richter scale is given by

$R\left(E\right)=\frac{2}{3}\log \left(\frac{E}{E_0}\right)$,

where $E_0$ is the energy of a certain small earthquakes. Compare the energies of the $1999$ and $1985$ earthquakes.

g. According to newspaper article, "Scientists say such an earthquake of magnitude $7.5$ could release $15$ times as much energy as the magnitude $6.7$ trembler that struck the Northbridge section of Los Angeles" in $1994.$Source: The New York Times. Using the formula from part f, verify this quote by computing the magnitude of an earthquake with $15$times the energy of a magnitude $6.7$ earthquake.