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Math Help - Earthquake Problem

  1. #1
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    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.
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  2. #2
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    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)
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  3. #3
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    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
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  4. #4
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    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.
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  5. #5
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    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?
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  6. #6
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    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.
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  7. #7
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    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?
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  8. #8
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    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.
    Last edited by badgerigar; December 6th 2007 at 03:31 PM. Reason: forgot \math tags
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