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 * Tank Example 1**

A tank initially contains 50 litres of water in which is disolved 10kg of salt. Brine containing salt with a concentration of 2kg/litre is flowing into the tank at a rate of 5 litres/minute. The mixture is stirred continuously and flows out of the tank at a rate of 3 litres/minute. Write the differential equation for the rate of change in the quantity of salt (**Q**) at time **t**.

From the question ... ... V(0) = 50 litres ... ... Q(0) = 10 kg

... ... C(in) = 2 kg/litre

math \\ . \qquad \dfrac{dV(in)}{dt} = 5 \; \text{litres/minute} \\. \\ . \qquad \dfrac{dV(out)}{dt} = 3 \; \text{litres/minute} math

Find Rate In ... ... **Rate In = (concentration of inflow) × (rate of inflow)**

math . \qquad \dfrac{dQ(in)}{dt} = C(in) \times \dfrac{dV(in)}{dt} math . math . \qquad \dfrac{dQ(in)}{dt} = 2 \times 5 = 10 \; \text{kg/minute} math

Find Volume at time t. math \\ . \qquad \dfrac{dV}{dt} = \dfrac{dV(in)}{dt} - \dfrac{dV(out)}{dt} \\ .\\ . \qquad \quad =5-3 . \qquad \quad =2 math

Integrate math \\ . \qquad V=\displaystyle{ \int 2 \; dt} \\. \\ . \qquad V=2t+c math

V(0) = 50, so c = 50

math . \qquad V=2t+50 math

Find Rate Out ... ... **Rate Out = (concentration within tank) × (rate of outflow)**

math . \qquad \dfrac{dQ(out)}{dt} = \dfrac{Q(t)}{V(t)} \times \dfrac{dV(out)}{dt} math . math \\ . \qquad \dfrac{dQ(out)}{dt} = \dfrac{Q}{2t+50} \times 3 \\. \\ . \qquad \qquad \quad \; = \dfrac{3Q}{2t+50} \; \text{kg/minute} math

Hence find the differential equation math . \qquad \dfrac{dQ}{dt} = \dfrac{dQ(in)}{dt} - \dfrac{dQ(out)}{dt} math . . math . \qquad \dfrac{dQ}{dt} = 10-\dfrac{3Q}{2t+50} \; \text{kg/minute} math

**Notice the DE contains two variables (Q and t) so we cant solve this using Year 12 calculus.**

We will examine methods of solving DEs like this using direction fields

Your calculator can solve most DEs like this. Go to the Interactive/Equations menu and select **dSolve**

Enter: Equation: **q'=10–3q/(2t+50)** inde var: **t** Depe var: **q** Condition: **t=0,q=10**
 * Include condition**