A tank is filled with 1000 liters of pure water. Brine containing 0.03 kg of sal

Question

A tank is filled with 1000 liters of pure water. Brine containing 0.03 kg of salt per liter enters the tank at 6 liters per minute. Another brine solution containing 0.09 kg of salt per liter enters the tank at 9 liters per minute. The contents of the tank are kept thoroughly mixed and the drains from the tank at 15 liters per minute. A. Determine the differential equation which describes this system. Let S(t) denote the number of kg of salt in the tank after t minutes. Then dS/dt = B. Solve the differential equation for S(t). S(t) =

Explanation / Answer

dS/dt = 0.18 + 0.81 - 15S/1000.
dS/dt = 0.99 - 0.015S.
I'm going to use a Laplace transform:
sL{S} - S(0) = 0.99/s - 0.015L{S}.
(s+0.015)L{S} = 0.99/s =>
L{S} = 0.99/[s*(s+0.015)].
Use partial fractions:
1/[s*(s+0.015)] = A/s + B/(s+0.015);
1 = As + 0.015A + Bs =>
A = 1/0.015 and B = -1/0.015.
L{S} = (0.99/0.015)/s - (0.99/0.015)/(s+0.015);
S(t) = 66*[1 - e^(-0.015t)].

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