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Math Help - Rate of evaporation from a dish fo water

  1. #1
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    Rate of evaporation from a dish fo water

    A dish has a shape described by the equation:
    h=(x^2+y^2)^3/2
    At time = 0 it is filled to a height of 20cm with a fluid that evaporates when exposed to air. The evaporation rate is proportional to the exposed surface area (that is decreasing) at any time t.
    if h(t) is the height of the fluid at time t then
    dh/dt is proportional to pir(t)^2, r(t) is the radius at time t. After 20 minutes the height of the fluid was 19.7cm.
    im trying to make a differential equation that governs the height h(t) during the evaporation.

    initially trying to write h as a function of r
    maybe
    dh = pi*r2 (t) dt???
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  2. #2
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    Re: Rate of evaporation from a dish fo water

    Hey azzarules.

    You already have given that dh/dt is proportional to pi*r(t)^2 so if you can express r(t) in terms of h(t) then you have dh/dt = A*pi*r(t)^2 and then solve for h(t) and use the boundary value (time = 20, height = 19.7) to solve for the constant A.

    What you had is right except proportionality means you need to include a constant (I've called it A in my post).
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  3. #3
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    Re: Rate of evaporation from a dish fo water

    so then we have

    dh= A*pi*r(t)^2 dt

    is the integration of this in terms of t
    A*pi*r(t)^2*t,
    or is this wrong? im being confused by the changing radius depending on time r(t)
    how can you solve this without knowing r(t)?
    Last edited by azzarules; November 6th 2012 at 11:17 PM.
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  4. #4
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    Re: Rate of evaporation from a dish fo water

    You don't need r and h in terms of t, before integrating. You have,

    \displaystyle \begin{align*} \frac{dh}{dt}\ &=\ A \pi r^2 &=\ A \pi h^{\frac{2}{3}} \end{align*}

    Pi can disappear into the unknown constant A...

    \displaystyle \begin{align*} \frac{dh}{dt}\ &=\ A h^{\frac{2}{3}} \end{align*}

    Then separate...

    \displaystyle \begin{align*}h^{\frac{3}{2}}\  \frac{dh}{dt}\ &=\ A \end{align*}


    Spoiler:

    Just in case a picture helps...



    ... where ...



    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case t), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).





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  5. #5
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    Re: Rate of evaporation from a dish fo water

    how can r^2 become h^2/3?


    dont worry worked i tout
    thanks
    Last edited by azzarules; November 7th 2012 at 01:50 PM.
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