An earthquake has a period of $\frac { 1 } { 2 }$ second, and an amplitude of $500,000$ micrometers. Find its measure on the Richter scale.
A) $3$
B) $2$
C) $5$
D) $6$
E) $8$

Question

Quizplus · Accepted Answer

Using the formula for Richter magnitude, we have: 
$$M_L = \log_{10} \left( \frac{A}{A_0} \right) + \log_{10} \left( \frac{1}{T} \right) +\dotsb$$ 
where $A$ is the amplitude of the earthquake, $T$ is the period, and $A_0$ is a reference amplitude. 
Plugging in the given values, we get: 
$$M_L = \log_{10} \left( \frac{500{,}000}{1} \right) + \log_{10} \left( \frac{1}{0.5} \right) +\dotsb$$ 
$$M_L = \log_{10} \left( 500{,}000 \right) + \log_{10} \left( 2 \right) +\dotsb$$ 
Using a calculator or logarithm table, we find that $$M_L \approx 6.3.$$ 
Rounding to the nearest integer, we get $\boxed{\textbf{(D) }6}$.

An Earthquake Has a Period Of 12\frac { 1 } { 2 }21​

An Earthquake Has a Period Of $\frac { 1 } { 2 }$