Thread: 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.)

-I have no idea how I should go about solving this problem. Please help.

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)

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

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 for some constant k.

--Kevin C.

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?

No. is the solution to .

For this one we have , and so


--Kevin C.

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

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

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.