Question

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.

A(t) = ____ lb

Answer 1

$${dA\over dt}=R_{in}-R_{out}$$

$$R_{in}=(2lb/gal)\cdot(5gal/min)=10lb/min$$

$$R_{out}=({A(t)\over 500}lb/gal)\cdot(5gal/min)={A(t)\over 100}lb/min$$

Hence

$${dA\over dt}=10-{A(t)\over 100}$$

$${dA\over 10-{A\over 100}}=dt$$ $$\ln\big|10-{A\over 100}\big|=-0.01t+\ln c$$

$$10-{A\over 100}=ce^{-{t\over 100}}$$

$$A=1000-100ce^{-{t\over 100}}$$

Apply the initial conditions

$$A(0)=0=1000-100ce^{-{0\over 100}}=>c=10$$

$$A(t)=1000-1000e^{-{t\over 100}}\ lb$$