A tank holds 10 liters of water. Brine containing 0.3 kg of…
A tank holds 10 liters of water. Brine containing 0.3 kg of salt per liter flows in at 2 L/min, mixes instantly, and the mixture drains out at the same rate. How much salt is in the tank after 5 minutes?
Explanation of Method
Balance the rate of salt entering against the rate leaving (proportional to the current concentration) to get a linear ODE for $Q(t)$, solve it, and evaluate at $t=5$.
Step 1: Set up the balance equation
Salt enters at $2\times0.3=0.6$ kg/min. Since volume stays constant at 10 L, salt leaves at $2\cdot\dfrac{Q}{10}=\dfrac{Q}{5}$ kg/min. Hence
\[\frac{dQ}{dt}=0.6-0.2Q,\qquad Q(0)=0.\]
Step 2: Solve the linear ODE
Separating, $\dfrac{dQ}{3-Q}=0.2\,dt$ (after factoring $0.6-0.2Q=0.2(3-Q)$), integrates to $3-Q=Ce^{-0.2t}$; with $Q(0)=0$, $C=3$:
\[\boxed{\,Q(t)=3\big(1-e^{-0.2t}\big)\ \text{kg}.}\]
Step 3: Evaluate at $t=5$
\[Q(5)=3(1-e^{-1})\approx3(1-0.3679)\approx1.90\ \text{kg}.\]
Result
Final answer:
\[\boxed{\,Q(5)=3(1-e^{-1})\approx1.90\ \text{kg}\qquad\big(Q(t)\to3\ \text{kg as } t\to\infty\big).}\]