A tank contains 10000 liters of a solution consisting of 100

A tank contains 10,000 liters of a solution consisting of 100 kg of salt dissolved in water. Pure rocky mountain spring water is pumped into the tank at the rate of 10 liters/second. (The mixture is kept uniform by stirring.) The mixture is pumped out of the tank at the same rate it enters. Set up the initial value problem for the amount of salt, S(t), in the tank at time t. Solve this problem for S(t). How long will it be before only 50 kg of salt remain in the tank?

Solution

ri = 10

ci=0

ro = 10

co(t) = x(t)/V(t)

V(t) = 1000 - (10-10)t = 1000

x(0) = 100

differential equation = rici -roco

= -10x/1000 = -x/100

=>

dx/x = -dt/200

ln x = -t/100 + C

x = C e^(-t/100)

x(0) = 100

=>

C = 100

x = 100e^(-t/100)

c)

50 = 100 e^(-t/100)

t = 69.32 seconds