Question

"On the Richter scale, the magnitude R of an earthquake can be measured by the intensity formula

R=log(I/Io)
where I is the intensity of the shock wave. Compare the intensity of an earthquake measuring 4.5 with the intensity of an earthquake measuring 3.0 on the Richter scale. Round your answer to the nearest whole number."
A) The earth quake measuring 4.5 on the Richter scale is about 1.8 times more intense than an earthquake measuring 3.
B) The earth quake measuring 4.5 on the Richter scale is about 32 times more intense than an earthquake measuring 3.
C) The earth quake measuring 4.5 on the Richter scale is about 9 times more intense than an earthquake measuring 3.
The earth quake measuring 4.5 on the Richter scale is about 63 times more intense than an earthquake measuring 3.

Answer 1

B) The earthquake measuring 4.5 on the Richter scale is about 32 times more intense than an earthquake measuring 3.

Explanation:

Let Ra = log(Ia/Io), and Rb = log(Ib/Io). Then for ...

Ra - Rb = 4.5 -3.0 = 1.5

log(Ia/Io) -log(Ib/Io) = 1.5 . . . . . . . . . . . substitute the given expressions

(log(Ia) -log(Io)) -(log(Ib) -log(Io)) = 1.5 . . . expand the logs

log(Ia) -log(Ib) = 1.5 . . . . simplify

log(Ia/Ib) = 1.5 . . . . . . . . write as a single log

Ia/Ib = 10^1.5 ≈ 32 . . . . take the antilog

The ratio of intensities is about 32, corresponding to choice B.