So the answer is there but I have no idea how the book got it. 1. Consider a tan

Question

So the answer is there but I have no idea how the book got it. 1. Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 150 li (L) of a dye solution with a concentration of 3 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 3 L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration ofdye in the tank reaches 2% of its original value ANSWER t= 50 In 50 ~ 195.6 min

Explanation / Answer

say amount of dye at any time t in grams = A(t)

A'(t) = inflow - outflow = 0*3 - (A(t)/150))*3 = 0 - (A(t)/150)*3 = -A(t)/50

=> A'(t) = -A(t)/50

=> dA/dt = -A/50 => dA/A = -dt/50

Integrate both sides

ln(A) = - t/50 + C where C is consant of integartion

A(0) = initial amount = 150*3 = 450g

ln(A(0)) = 0 + C => C = ln(450)

=> ln(A) = - t/50+ ln(450)

=> ln(A/450) = -t/50

when concentration of dye is 2% of original , A(t) = 2% of 450g = 2/100 * 450 = 9g

so , ln(9/450) = -t/50

=> -ln(50) = -t/50

=> t = 50 ln(50) Answer

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