A tank contains 90 kg of salt and 1000 L of water. A solution of a concentration 0.045 kg of salt per liter enters a tank at the rate 5 L/min.

Question

A tank contains  90kg of salt and  1000L of water. A solution of a concentration 0.045 kg of salt per liter enters a tank at the rate 5 L/min. The solution is mixed and drains from the tank at the same rate.

a)What is the concentration of our solution in the tank initially?

my response is 0.09Kg/L (Which is correct)

b) Find the amount of salt in the tank after 2 hours.

my response is 89.55kg (Which is incorrect)

c)Find the concentration of salt in the solution in the tank as time approaches infinity.

my response is 0.045Kg/L (which is correct)

I need help for the answer in part B

Answer 1

Let Q(t) is mass of solt in tank in moment t.

Then Q'(t) = 0.045kg/L· 5L/min - Q(t)/1000·5;or Q' = 0.225 - Q/200;

Q' + Q/200 = 0.225; dQ/dt = 0.225 - Q/200; dQ/(0.225 - Q/200) = dt;∫

∫dQ/(0.225 - Q/200) = ∫dt; - 200ln(Q/200 - 0.225) = t - 200 lnC; Q/200 - 0.225 = Ce^-t/200. At t = 0 Q(0) = 90

So, 90/200 - 0.225 = C, C = 0.225 and now Q(t) = 45 + 45e^{- t/200}_;

a) Initial concentration 90kg/1000L = 0.09 kg/L

b) Q(120) = 45 + 45e^{- 120/200}= 69.70 kg

c) limQ(t) = 45 kg and concentration of solt is 45/1000 = 0.045 kg/L

t →∞