Suppose that a large mixing tank initially holds 300 gallons

Suppose that a large mixing tank initially holds 300 gallons of water in which 30 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0. What is A(0)? (Use A for - A(0)? (Use A d for A(t))

Solution

A(0)=30 pounds

Inflow and outflow rates are the same.Hence the net volume remains unchanged at 300 gallons

dA=-(A/300)*3dt=-Adt/100

dA/dt=-A/100

dA/A=-dt/100

Integrating gives

ln(A)=-t/100

A=Ce^{-t/100}

A(0)=30

Hence, C=30

A=30 e^{-t/100}