Question

The Richter scale measures the intensity of earthquakes on a logarithmic scale. The magnitude of an earthquake on the Richter scale can be defined by M=2/3log(E)-3.2, where E is the energy of the quake in joules. The 2011 Tohoku earthquake in Japan measured 9.1 on the Richter scale. The 1999 Hector Mine earthquake in eastern California had a magnitude of 7.1. Calculate the energy released by each earthquake.

Answer 1

Japan :
`E=2.8183829313* 10^(18)`

California :
`E=2.8183829313* 10^(15)`

Explanation:

The magnitude of an earthquake on the Richter scale can be defined by

`M=(2)/(3)\log (E)-3.2`

where E is the energy of the quake in joules.

Add 3.2 on both sides.

`M+3.2=(2)/(3)\log (E)`

Multiply both sides by 3/2.

`(3)/(2)(M+3.2)=(3)/(2)* (2)/(3)\log (E)`

`1.5M+4.8=\log (E)`

`10^(1.5M+4.8)=E`
`[\because \log x=a\Rightarrow x=10^a]`

`E=10^(1.5M+4.8)` .... (1)

It is given that the 2011 Tohoku earthquake in Japan measured 9.1 on the Richter scale.

Substitute M=9.1 in equation (1).

`E=10^(1.5(9.1)+4.8)`

`E=10^(18.45)`

`E=2.8183829313* 10^(18)`

Therefore, the energy released by earthquake in japan is
`E=2.8183829313* 10^(18)`.

It is given that the 1999 Hector Mine earthquake in eastern California had a magnitude of 7.1.

Substitute M=7.1 in equation (1).

`E=10^(1.5(7.1)+4.8)`

`E=10^(15.45)`

`E=2.8183829313* 10^(15)`

Therefore, the energy released by earthquake in California is
`E=2.8183829313* 10^(15)`.