Question

The Richter Scale gives a mathematical model for the magnitude of an earthquake in terms of the ratio of the amplitude of an earthquake wave (also known as the intensity of an earthquake) to the amplitude of the smallest detectable wave. If we denote the magnitude of the earthquake by R, the intensity (also known as the amplitude of the earthquake wave) of the earthquake by A, and the amplitude of the smallest detectable wave by S, then we can model these values by the formula

R=log(A/S)

In 1906, San Francisco felt the impact of an earthquake with a magnitude of 7.8. Many pundits claim that the worst is yet to come, with an earthquake 451,710 times as intense as the 1906 earthquake ready to hit San Francisco. If the pundits ability to predict such earthquakes were correct, what would be the magnitude of their claimed earthquake? Round your answer to the nearest tenth.

Answer 1

Let the 1906 earthquake's intensity was A₁ and magnitude was R₁.

Also let the later earthquake's magnitude is R₂ and intensity is A₂​​​​​​

Then we have,

R₁ = log(A₁/S) and R₂ = log(A₂/S)

.

So, R₂ - R₁ = log(A₂/S) - log(A₁/S)

i.e. R₂ - R₁ = log(A₂/A₁)

.

Now according to the given data, we have,

R₁ = 7.8 and A₂/A₁ = 451710

.

So we have,

R₂ - 7.8 = log(451710)

i.e. R₂ = 7.8 + log(451710)

i.e. R₂ ≈ 7.8 + 5.7

i.e. R₂ ≈ 13.5

.

So the claimed earthquake will have a magnitude of 13.5