Mixing Problem A tank initially holds 40 litres of water in which 1 kg of salt h

Question

Mixing Problem A tank initially holds 40 litres of water in which 1 kg of salt has been dissolved. Brine containing 0.2 kg of salt per litre enters the tank at 8 litres/min and the well-stirred mixture leaves the tank at the rate of 12 litres per minute.

a) Let y(t) denote the amount of salt in the tank at time t. Show that dy/dt=1.6-(3y/10-t)

b) Find the amount of salt in the tank at time t.

c) Find the amount of salt in the tank after 10 minutes.

d) Use Maple to plot the graph of y(t) . At what time is the amount of salt in the tank the greatest? How much salt is in the tank at that time?

Explanation / Answer

dy/dt=1.6-(3y/10-t)

dy/dt-1.6=-(3y/10-t)

dy/dt+(3y/10-t)=1.6

which is linear differential equation in which integrating factor=e^integral(1/(10-t)=e^(-log(t-10)=1/(t-10)

so solution is y/(t-10)=integral(1.6/(t-10) dt+c

y/(t-10)=1.6log(t-10)+c

intiallly at t=0, y=0.2

0.2/(-10)=1.6log(10)+c

c=-0.2/10-1.6log(10)

so  y/(t-10)=1.6log(t-10)+-0.2/10-1.6log(10)

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