A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt
in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/min, and
the mixture is allowed to flow out of the tank at a rate of 2 gal/min. Find the amount
of salt in the tank at any time prior to the instant when the solution begins to overflow.
Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of
overflowing. Compare this concentration with the theoretical limiting concentration if the
tank had infinite capacity.
2. The attempt at a solution
I get stuck in the very beginning where you have to set up the differential equation.
dQ/dt = rate of flow in - rate of flow out
rate of flow in = 3gal/min × 1 lb/gal = 3 lb/min
(C = concentration, Q = quantity of salt, V = Volume)
rate of flow out = C r = (Q / V) r
The volume changes; starts at 200 and 3-2=1 gallon per minute extra:
V = 200 + t
So: dQ/dt = 3 - (Q / 200+t)2, which I would then have to solve.
But according to my answer book the equation that has to be solved is:
dQ/dt = 3 - Q(t)
Does anyone understand how they get there?