A tank contains $500$ litres of brine in which $75$ kg of salt is dissolved. Brine containing \(

Question

A tank contains $500$ litres of brine in which $75$ kg of salt is dissolved. Brine containing $3$ kg of salt per $10$ litres of water flows in at a rate of $20$ litres per minute, and the mixture flows out at the same rate. Show that, for $t \ge 0, \frac{\mathrm{d}x}{\mathrm{d}t} = \frac{150 - x}{25}$, where $x$ is the amount of dissolved salt in the tank at time $t \ge 0$. When will there be $125$ kg of dissolved salt in the tank? How much dissolved salt is in the tank after a long time? State an assumption used in the model.

Accepted Answer

(i) $\dfrac{\mathrm dx}{\mathrm dt}=\dfrac{150-x}{25}$ (ii) $t=25\ln3\approx27.5\text{ min}$ (iii) $150\text{ kg}$ (iv) The contents are kept thoroughly and uniformly mixed

Differential Equations - Mixing and Concentration Models

Differential Equations - Mixing and Concentration Models

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