initial amount of salt = "10m^3"
rate of pumping in = 200L/min
Let A be amount of salt at any time t,
then
"\\frac{dA}{dt} = R_{in} - R_{out}"
R is rate of flow.
"R_{in} = \\frac{1g}{1L} \\frac{5L}{1 min} = 5 g\/min"
"R_{out} = \\frac{A}{200}*{5} = \\frac{A}{40} g\/min"
Hence equation of state will be,
"\\frac{dA}{dt} = 5-\\frac{A}{40}"
solving equation,
"\\frac{dA}{dt} + \\frac{1}{40}A =5"
"e^{\\frac{1}{40}t}A = \\int 5e^{\\frac{1}{40}t}dt"
"e^{\\frac{1}{40}t}A = 200e^{\\frac{1}{40}t} + C"
"A = 200 + Ce^{-\\frac{1}{40}t}"
applying condition, A=20 at t=0
"20 = 200 + C \\implies C = -180"
Hence desired equation of the state is given by,
"A = 200 - 180e^{-\\frac{1}{40}t}"
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