Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 L of a dye solution with a concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water ﬂowing in at a rate of 2 L/min, the well-stirred solution ﬂowing out at the same rate.

a. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.

Question

Emerson Henderson

Mathematics

3 February, 18:04

Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 L of a dye solution with a concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water ﬂowing in at a rate of 2 L/min, the well-stirred solution ﬂowing out at the same rate.

a. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.

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Answers (1)

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Vincent Weber · Answer 1

Answer: t = 460.52 minutes

Step-by-step explanation:Q'=Q/100

Q' = rate in and out of water

Finding the differential equation

Let Q' (t) = The quantity of dye in the tank for t time

But rate in=0 Q/200 * 2=Q'

Q'/Q=-1/100

Dividing by Q gives

Ln/Q / + c = - 1/100 + c1

Integrating both sides gives

Ln/Q / = - (1/100) t + c2

But c+c1=C2 = A constant

Q=C2e (-t/100)

200e - (t/100)

t = ln200

t=460.52minutes