A 75 gallon capacity tank initially contains 50 gallons of f

A 75 gallon capacity tank initially contains 50 gallons of fresh water. Brine (a mixture of water and salt) at a concentration of 2 lbs/gal starts to run into the tank at 3 gal/min. At the same time, the well-stirred mixture of fresh water and brine runs out at 2 gal/min. Write an initial value problem (differential equation plus initial condition) that describes the rate of change of salt S in the tank at any time t. You do not need to solve the VP.

Solution

inflow rate is 3 gal/min and outflow is 2 gal/min

And initial volume of solution is 50 gallon

SO volume at time t minutes is :V=50+t

But maximum volume is 75 gallons. So t<=25

For t>25, Volume, V=75

Case 1:t<=25

dS=(2 lbs/gal)(3 gal/min)dt min-(S/V)2dt

dS=6dt-2Sdt/(50+t)

dS/dt=6-2S/(50+t),S(0)=0

Case 2:t>25

dS=6dt-2Sdt/75

dS/dt=6-2S/75

Initial condition for this case is S(25) is obtained using the solution from case 1