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Math Help - Linear ODE's

  1. #1
    Super Member Aryth's Avatar
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    Linear ODE's

    A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius \frac{1}{2} cm has radius 0.4 cm after 6 months, how long will it take:

    (a) For the radius to be \frac{1}{4} cm?

    (b) For the volume of the mothball to be half of what it was originally?
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  2. #2
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    Quote Originally Posted by Aryth View Post
    A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius \frac{1}{2} cm has radius 0.4 cm after 6 months, how long will it take:

    (a) For the radius to be \frac{1}{4} cm?

    (b) For the volume of the mothball to be half of what it was originally?
    Here's a start: \frac{dV}{dt} = -k \pi r^2 = -k \pi \left(\frac{3V}{4 \pi}\right)^{2/3}.
    Last edited by mr fantastic; October 23rd 2009 at 04:11 AM. Reason: Dropped a negative
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by mr fantastic View Post
    Here's a start: \frac{dV}{dt} = k \pi r^2 = k \pi \left(\frac{3V}{4 \pi}\right)^{2/3}.
    The title of the thread is : Linear ODE...

    ... and that could mean that a linear ODE is required. This goal is achieved writing...

    \frac{dV}{dt} = 4\cdot \pi \cdot r^{2} \frac{dr}{dt} = -4\cdot \pi \cdot r^{2} \cdot \alpha \rightarrow \frac{dr}{dt} = - \alpha (1)

    Very easy!...

    Kind regards

    \chi \sigma
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  4. #4
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    Quote Originally Posted by chisigma View Post
    The title of the thread is : Linear ODE...

    ... and that could mean that a linear ODE is required. This goal is achieved writing...

    \frac{dV}{dt} = 4\cdot \pi \cdot r^{2} \frac{dr}{dt} = -4\cdot \pi \cdot r^{2} \cdot \alpha \rightarrow \frac{dr}{dt} = - \alpha (1)

    Very easy!...

    Kind regards

    \chi \sigma
    My DE has the form \frac{dt}{dV} = f(V). Last time I checked, such DE's were classified as linear. However, I do agree that your DE \frac{dr}{dt} = - \alpha provides for a much simpler solution.
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  5. #5
    Super Member Aryth's Avatar
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    Thanks a lot. The word problems really mess me up.
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