# Earthquake Problem

• Dec 5th 2007, 07:17 PM
blurain
Earthquake Problem
1) Earthquake magnitude M, as measured on the Richter scale, increases at a rate proportional to the reciprocal of x, where x is the normalized seismograph reading, measured in millimeters.

a) Write the equation for M'(x)
b) Find a general formula for M(x). (The formula should contain two undetermined constants.)

• Dec 5th 2007, 10:32 PM
Quote:

M ... increases at a rate proportional to the reciprocal of x,
All you have to do for part a) is think about what this phrase means. After that you can have a shot at solving b)
• Dec 6th 2007, 06:24 AM
blurain
Hmm, okay, the reciprocal of x would be x^-1, or 1/x...but I am confused with the statement "increases at a rate proportional to..."

Does this mean that dM/dT= M(1/x)? I don't know if this is right :confused:
• Dec 6th 2007, 09:45 AM
TwistedOne151
Proportionality
Remember, when you say that "y is proportional to x", you are saying that y=kx, for some constant k (called the constant of proportionality). So if the rate of increase of M is proportional to the inverse of x, it means $\displaystyle \frac{dM}{dx} = \frac{k}{x}$ for some constant k.

--Kevin C.
• Dec 6th 2007, 10:40 AM
blurain
So if M'(x) = k/x, does this mean that the general formula for M(x), which would contain two undetermined constants, have to use Ae^kt to figure out?
• Dec 6th 2007, 10:48 AM
TwistedOne151
Not exactly
No. $\displaystyle Ae^{kx}$ is the solution to $\displaystyle M'(x)=kM$.

For this one we have $\displaystyle M'(x)=\frac{k}{x}$, and so
$\displaystyle M(x)= \int \frac{k}{x} \,dx = k \int \frac{dx}{x}$

--Kevin C.
• Dec 6th 2007, 11:05 AM
blurain
Ah, I see. Then if M(1)=3, I would be writing it as $\displaystyle 3= k \int \$1 (dx)?

And if so, how would I solve for one of the constants?
• Dec 6th 2007, 02:28 PM