Chapter 4.5, Problem 65E

Richter Scale The Richter scale provides a measure of the magnitude of an earthquake. In fact, the largest Richter number $M$ ever recorded for an earthquake was $8.9$ from the 1933 earthquake in Japan. The following formula shows a relationship between the amount of energy released and the Richter number.

$M=\frac{2}{3}\log \frac{E}{0.007},$

Where $E$ is measured in kilowatt-hours. Source: Mathematics Teacher.

a. For the 1933 earthquake in Japan, what value of $E$ gives a Richter number $M=8.9?$.

b. If the average household uses 247 kWh per month, how many months would the energy released by an earthquake of this magnitude power 10 million household?

c. Find the rate of change of the Richter number $M$ with respect to energy when $E=\mathrm{70,000}$ kWh.

d. What happens to $dM/dE$ as $E$ increases?