Modeling with First Order DE’s — Question 1

Question 1 – Solution

Strategy. Write separate liquid and salt balances, then use the quotient rule to model concentration in a changing volume.

Step 1: Balance volume and salt. The volume rate is $V'=3-1=2$ L/min, so $V=100+2t$ L. Capacity is reached at $t=50$ min. Up to that time, $$S'=0.60-\frac{S}{100+2t},\qquad S(0)=10.$$ Both terms have units kg/min: salt enters at $(3)(0.20)$ and leaves at $(1)(S/V)$. The pre-overflow model applies for $0\le t<50$, with a continuous value at $50$.

Step 2: Derive the concentration equation. Since $S=cV$, $S'=Vc'+cV'$. Hence $$(100+2t)c'=0.60-3c,\qquad c(0)=0.10.$$ Using the integrating factor $(100+2t)^{3/2}$ and $c(0)=0.10$ gives $$\boxed{c(t)=0.20-0.10(100/(100+2t))^{3/2}}.$$ Indeed $c'=3(0.20-c)/V$, verifying the concentration balance. Omitting $cV'$ would ignore the dilution caused by increasing volume.

Step 3: Locate the target and capacity values. The concentration increases strictly. Setting $c=0.15$ gives $$\boxed{t_*=50(2^{2/3}-1)\approx 29.37\text{ min}<50\text{ min}}.$$ At capacity, $c(50)=0.20-0.10/2^{3/2}$, so $$\boxed{S(50)=200c(50)=40-5\sqrt 2\approx 32.93\text{ kg}}.$$

Step 4: Respect the operating boundary. The final point is the instant the tank becomes full. Beyond it, continued inflow would require overflow or a changed flow rate. Either changes the volume and salt balances, so the formula is not a model of the same operation after $50$ min.