HardMathematicsHigh School (11–12)Functions and CalculusNumerical

A small chemical plant keeps a tank that holds 100 litres of liquid and 10 kg of dissolved salt at…

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A small chemical plant keeps a tank that holds $100$ litres of liquid and $10\ \text{kg}$ of dissolved salt at the start. Brine of $0.3\ \text{kg}$ of salt in each litre flows in at $5$ litres a minute. The stirred mixture flows out at $3$ litres a minute, so the tank slowly fills. The engineer writes the volume as $V = 100 + 2t$ litres. She sets up the equation $\dfrac{dQ}{dt} = 5 \times 0.3 - 3\dfrac{Q}{V}$ and solves it with an integrating factor, getting $Q = 0.3V - \dfrac{20000}{V^{3/2}}$ kilograms, where the constant is fixed by the starting salt. She wants the state of the tank $50$ minutes after the start. Find the amount of salt in kilograms, the volume in litres and the concentration of the liquid in kilograms per litre.

Need a reminder? The key ideaShow

Newton's law is $\dfrac{dT}{dt} = -k(T - T_s)$ , so $T = T_s + (T_0 - T_s)e^{-kt}$ . For a tank, $\dfrac{dQ}{dt} = r_{\text{in}}c_{\text{in}} - r_{\text{out}}\dfrac{Q}{V}$ , and the volume is $V = V_0 + (r_{\text{in}} - r_{\text{out}})t$ .

\displaystyle T - T_s = (T_0 - T_s)e^{-kt}.

The worked solution

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