A tank at the soda factory initially contains 1000 gallons of soda, with a sugar

Question

A tank at the soda factory initially contains 1000 gallons of soda, with a sugar concentration of .2 pounds of sugar per gallon of soda. Starting at time 0, soda with a concentration of .4 pounds of sugar per gallon of soda is pumped into the tank at a rate of 10 gallons per minute, and soda is pumped out of the tank at the same rate. Assuming instantaneous mixing, find the total amount of sugar in the tank at time t minutes. At what time does the concentration of sugar in the tank reach .3 pounds per gallon?

Explanation / Answer

If there's .01kg/liter entering, at 3L/min,that's .03kg/minute. That's important :). dS/dt = .03 - D (The rate at which the sugar level (S) changes per minute (t) = the rate at which sugar enters minus the rate at which it leaves) D= kS ( The drainage rate is proportional to the amount of sugar) So, substituting: dS/dt=.03 - kS Now, get dt alone: dS =(.03 -kS)dt dS/(.03-kS)=dt 1/(.03-kS) * dX = dt Now: INTEGRATE :) -ln | .03 -kS | + C1 = t + C2 (Each side has a different arbitrary C) subtracting the arbitrary C1 from both sides, we get a new arbitrary C on the right: -ln|.03-kS|=t+C3 ln|.03-kS|=-t-C4

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