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KillerZ
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Homework Statement
A tank contains 200 liters of fluid which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.
Homework Equations
initial amount of liquid = 200L
A(t=0) = 30g
rate in = 4L/min
rate out = 4L/min
concentration of the inflow liquid = 1 g/L
The Attempt at a Solution
[tex]\frac{dA}{dt} = (1g/L)(4L/min) - \frac{(4L/min)(A(t))g}{(200L) + (4L/min - 4L/min)t}[/tex]
[tex]\frac{dA}{dt} = (4) - \frac{4A}{200}[/tex]
[tex]\frac{dA}{dt} + \frac{4A}{200} = (4)[/tex]
[tex]\frac{dA}{dt} + \frac{A}{50} = (4)[/tex]
[tex]I(t) = e^{\int\frac{A}{50}dt}[/tex]
[tex]I(t) = e^{\frac{At}{50}}[/tex]
[tex]f(t) = \frac{1}{e^{\frac{At}{50}}}[\int (e^{\frac{At}{50}})(4) dt + c][/tex]
[tex]f(t) = \frac{1}{e^{\frac{At}{50}}}[\frac{200e^{\frac{At}{50}}}{A} + c][/tex]
[tex]f(t) = \frac{200e^{\frac{At}{50}}}{Ae^{\frac{At}{50}}} + \frac{c}{e^{\frac{At}{50}}}][/tex]