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

Let Q be the amount of sait in the tank and "dt\/dQ"

be the rate of change of amount in the tank

We get the equation as

"\\space dt\/\ndQ\n\u200b\t\n + \n1\/20\n\n\u200b\t\n Q=0"

It is linear equation and the solution of it is "Q=Ce ^{-\\frac{t}{20}\nt}\n\u200b"

At t=0 , the value of Q is 20

"\u21d2C=20"

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

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