1. 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.)

2. 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)

3. 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

4. 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 $\frac{dM}{dx} = \frac{k}{x}$ for some constant k.

--Kevin C.

5. 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?

6. Not exactly

No. $Ae^{kx}$ is the solution to $M'(x)=kM$.

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

--Kevin C.

7. Ah, I see. Then if M(1)=3, I would be writing it as $3= k \int \$1 (dx)?

And if so, how would I solve for one of the constants?

8. Ah, I see. Then if M(1)=3, I would be writing it as 1 (dx)?
No. You need to perform the integration before you substitute in (although writing this is making me wonder why. oh well)

M = [tex]k\int(1/x)dx[\MATH]
M = k log|x| + c
then you can substitute and I am sure you will be able to solve for one of the values.