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Thread: Differntial Caculus Salt Problem

  1. #1
    Member zangestu888's Avatar
    Jan 2009

    Differntial Caculus Salt Problem

    A resevoir contains 700 liters of pure water. Brine contaning 0.02 kg/L of salt enters at a rate of 5 L/min. Another source of brine contanining 0.05 kg/L enters at a rate of 2L/min. The resevoir is well mixed and drains at a rate of 7L/min.How much salt is thier in the resevoir (i) after t min (ii) after 10 min?

    Can anyone please help me start of this problem much thanks!
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  2. #2
    Senior Member
    Dec 2008
    In our problem, the level of water remains the same, as we have 5\frac{\mbox{L}}{\mbox{min}} and 2\frac{\mbox{L}}{\mbox{min}} flowing in and 7\frac{\mbox{L}}{\mbox{min}} flowing out.

    If we call the salt level S(t), then we have


    To find \frac{dS}{dt}, we subtract the rate of salt flowing out from the rate flowing in:

    \frac{dS}{dt}=\frac{dS_{\mbox{\scriptsize{in}}}}{d  t}-\frac{dS_{\mbox{\scriptsize{out}}}}{dt}.

    The rate flowing in is

    <br />
\begin{aligned}<br />
\frac{dS_{\scriptsize{\mbox{in}}}}{dt}&=0.02\frac{  \mbox{kg}}{\mbox{L}}\cdot 5\frac{\mbox{L}}{\mbox{min}}+0.05\frac{\mbox{kg}}{  \mbox{L}}\cdot 2\frac{\mbox{L}}{\mbox{min}} \\<br />
&=0.1\frac{\mbox{kg}}{\mbox{min}}+0.1\frac{\mbox{k  g}}{\mbox{min}}\\<br />
&=0.2\frac{\mbox{kg}}{\mbox{min}}.<br />
\end{aligned}<br />

    Since the fluid is well-mixed, the rate flowing out will be in proportion to 7\,\mbox{L} as the current salt level is to the entire resevoir. Therefore,

    <br />
\begin{aligned}<br />
\frac{dS_{\scriptsize{\mbox{out}}}}{dt}&=\frac{S}{  700\,\mbox{L}}\cdot 7\frac{\mbox{L}}{\mbox{min}} \\<br />
&=\frac{S}{100}\,\mbox{min}^{-1}.\\<br />
\end{aligned}<br />

    For \frac{dS}{dt}, we obtain

    <br />
\begin{aligned}<br />
\frac{dS}{dt}&=\frac{dS_{\mbox{\scriptsize{in}}}}{  dt}-\frac{dS_{\mbox{\scriptsize{out}}}}{dt}\\<br />
&=0.2\frac{\mbox{kg}}{\mbox{min}}-\frac{S}{100}\,\mbox{min}^{-1}.<br />
\end{aligned}<br />

    Now, all that remains is to solve the differential equation using the initial conditions given.
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