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Math Help - Differential Equation Problem

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    Differential Equation Problem

    A spherical mothball evaporates uniformly at a rate proportional to its surface area. Hence deduce a differential equation that links its radius with time. Given that the radius halves in one month, how long will the mothball last?

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  2. #2
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    Quote Originally Posted by eugenius View Post
    A spherical mothball evaporates uniformly at a rate proportional to its surface area. Hence deduce a differential equation that links its radius with time. Given that the radius halves in one month, how long will the mothball last?

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    A spherical mothball evaporates uniformly at a rate proportional to its surface area.

    \frac{dV}{dt} = -kS

    since V = \frac{4}{3}\pi r^3

    \frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt} = S \cdot \frac{dr}{dt}

    equating ...

    S \cdot \frac{dr}{dt} = -kS

    \frac{dr}{dt} = -k

    so, the radius decreases at a constant rate ... can you take it from here?
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    thanks, i'll see how i get on
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  4. #4
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    i still cant do the last bit, help please :S
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  5. #5
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    Quote Originally Posted by eugenius View Post
    i still cant do the last bit, help please :S
    \frac{dr}{dt} = -k \Rightarrow r = -kt + C.

    When t = 0, r = r_0. Therefore r = r_0 - kt.

    When t = 1, r = \frac{r_0}{2} \Rightarrow k = \frac{r_0}{2}.

    Therefore r = r_0 \left(1 - \frac{t}{2} \right).
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