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Math Help - help with creating differential equation..

  1. #1
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    help with creating differential equation..

    not understanding.

    A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.

    Do i need to use the equations for volume of a sphere and surface area?


    I got this:
    \frac{d(4/3)(pi)r^3}{dt} = 4pi(r^2) -  \frac{4}{3}pi(r^3)

    I think this is highly wrong tho.
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  2. #2
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    Quote Originally Posted by p00ndawg View Post
    not understanding.

    A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.

    Do i need to use the equations for volume of a sphere and surface area?


    I got this:
    \frac{d(4/3)(pi)r^3}{dt} = 4pi(r^2) - \frac{4}{3}pi(r^3)

    I think this is highly wrong tho.
    Close though. Let A be the surface area and V the volution. Your told that

    \frac{dV}{dt} = k A or using the usual formula's (as you did)

    \frac{4 \pi}{3} \frac{d r^3}{dt} = k 4 \pi r^2\;\;\;\Rightarrow\;\;\;\frac{4 \pi}{3} 3 r^2 \frac{d r}{dt} = k 4 \pi r^2

    or

     \frac{d r}{dt} = 3k .

    That's a ODE for the radius. You want one for the volume. Let's go back to

    \frac{dV}{dt} = k A

    Since V = \frac{4}{3} \pi r^3\;\;\;\text{then}\;\;\;r = \left( \frac{3V}{4\pi} \right) ^{1/3}. Since  A = 4 \pi r^2 = 4 \pi \left( \frac{3V}{4\pi} \right) ^{2/3}

    then

    \frac{dV}{dt} = k V^{2/3}

    where I absorded all of the numbers in the k.
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