Alarge 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.
There's no salt in the tank at the start, so A(0) = 0.
Salt flows in at a rate of
(2 lb/gal) * (5 gal/min) = 10 lb/min
and flows out at a rate of
(A(t)/500 lb/gal) * (5 gal/min) = A(t)/100 lb/min
Then the net flow rate is given by
A'(t) = 10 - A(t)/100
Solve the ODE:
With A(0) = 0, the above gives us 0 = 1000 + C, so that C = -1000, and the particular solution to this IVP is