COMMENTS/RESPONSE MATHS - 6.EQUATIONS

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Can you help me with this problem? A tank with a capacity of 500gallons originally containing 200 gallons of water with 50 pounds of salt in the solution. Water containing 2 pounds of salt per gallon is entering at a rate of 3 gallons per minute, and the mixture is allowed to flow out of the tank at a rate of 2 gallons per minute. Find the amount of salt in the tank at anytime t prior to the moment when the solution begins to overflow from the tank. Find the concentration (in pounds per gallon) of salt at the moment the tank begins to overflow. Compare this concentration with the theoretical limiting concentration if the tank had infinite capacity. Thanks

LET US FORMULATE THE PROBLEM
INITIAL SOLUTION IN TANK = 200 G
INITIAL SALT IN TANK = 50 LB.
INFLOW = 3 GPM..
WITH SALT CONTENT = 2 LB/G
OUT FLOW = 2 GPM.
NET INFLOW = 3-2 = 1 GPM

LET AT ANY TIME ....T......,MINUTES
THE SALT IN TANK BE S LB.
SOLUTION IN TANK 200 + T*1= (200+T) G
SALT CONCN. = S / (200+T) LB/G
AFTER DT M.
INFLOW = 3DT G
SALT IN = 3DT*2 = 6DT LB.
OUT FLOW = 2DT G
SALT OUT = 2DT*S / (200+T) LB.
CHANGE IN SALT CONTENT DURING DT IS
DS = SALT IN - SALT OUT
DS = 6DT - 2SDT / (200+T)
DS/DT = [1200+6T-2S] / (200+T)
THIS IS A STANDARD NON HOMOGENIOUS EQN.WHICH YOU CAN CONVERT INTO
HOMOGENEOUS EQN. BY PUTTING
200+T = U ...

SO THAT YOU GET A HOMOGENEOUS
EQN.....WHICH YOU CAN SOLVE BY SUBSTITUTION
S = V U
THE CONSTANT OF INTEGRATION CAN BE OBTAINED FROM S = 50 WHEN T=0
THIS WILL GIVE YOU A GENERAL EQN FOR SALT CONTENT TILL THE TANK GETS
FULL WHICH NEEDS (500-200) / 1 = 300 MTS.AS
1 GPM IS THE NET INFLOW
HENCE YOU CAN GET SALT CONTENT
AT T = 300 TO FIND THE ANSWER WHEN TANK IS ABOUT TO OVER FLOW.
TO GET CONCN.DIVIDE S BY TOTAL SOLUTION.

= S / (200+T)
OBVIOUSLY IF THE TANK HAS INFINITE CAPACITY, IT WILL EQUAL THE
CONCENTRATION OF INCOMING SOLUTION
THAT IS 2 LB/GALLON.