A tank initially contains $2000$ litres of water and $20$ kg of dissolved salt. Brine with $C$

Question

A tank initially contains $2000$ litres of water and $20$ kg of dissolved salt. Brine with $C$ kg of salt per $1000$ litres is entering the tank at $5$ litres per minute. The amount of salt in the tank at time $t$ minutes is $x$ kg. Assume that the solution is always uniformly mixed. Show that $\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{1}{400}\left( 2C-x \right)$. Hence determine the value of $C$ if the amount of salt in the tank remains constant at $120$ kg after certain time has passed.[3] Find $x$ in terms of $t$.[5] Sketch the graph of the particular solution, including the coordinates of the point(s) where the graph crosses the axes and the equations of any asymptotes. Find the time $t$ when the amount of salt in the tank is $60$ kg, giving your answer to the nearest minute.[3] State one assumption for the above model to be valid.[1]

Accepted Answer

(i) $C=60\text{ kg per }1000\text{ L}$ (ii) $x=20(6-5\mathrm e^{-t/400})$ (iii) Intercept $(0,20)$, asymptote $x=120$, $t=204\text{ min}$ when $x=60$ (iv) For example, no evaporation

2019 YIJC P1 Q10

2019 YIJC P1 Q10

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