Question #244047

Situation 2

A tank initially contains 100 gal of brine where there is dissolved 20lb of salt. Starting at time t = 0, a

brine containing 3lb of dissolved salt per gallon flows into the tank at the rate of 6gal/min is kept uniform

by stirring and the well-stirred mixture simultaneously flows out of the tank at a slower rate of 3gal/min.

1. How much salt is present at the end of 15 min?

2. How much salt is present after a long time?

A tank initially contains 100 gal of brine where there is dissolved 20lb of salt. Starting at time t = 0, a

brine containing 3lb of dissolved salt per gallon flows into the tank at the rate of 6gal/min is kept uniform

by stirring and the well-stirred mixture simultaneously flows out of the tank at a slower rate of 3gal/min.

1. How much salt is present at the end of 15 min?

2. How much salt is present after a long time?

Expert's answer

LetÂ QÂ be the amount of sait in the tank andÂ dt/dQ

â€‹Â be the rate of change of amount in the tank

We get the equation asÂ

\space dt/ dQ â€‹ + 1/20 â€‹ Q=0

It is linear equation and the solution of it isÂ Q=Ce ^{-\frac{t}{20} t} â€‹

AtÂ t=0Â , the value ofÂ QÂ isÂ 20Â

â‡’C=20

Therefore the equation is\space Q=Ce ^{-\frac{t}{20} t}

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