Mixing Tank Problems:
1. A tank contains 100 gal of pure water. A sugar-water
solution containing 0.4 lb of sugar per gal enters the tank at a rate of
3 gal per minute and simultaneously a drain is opened at the bottom of
the tank allowing the well stirred sugar solution to leave at 3 gal per
minute.
(i) What will be the sugar content in the tank after 25 minutes?
(ii) How long will it take the sugar content in the
tank to reach 30 lb?
(iii) What will be the eventual ( \(t \to
\infty\) ) sugar content in the tank?
2. A tank with a capacity of 500 gal originally contains
200 gal of water with 100 lb of salt in solution. Water containing 1 lb
of salt per gallon is entering at a rate of 3 gal/min, and the mixture
is allowed to flow out of the tank at a rate of 2 gal/min. Find the
amount of salt in the tank at any time prior to the instant when the
solution begins to overflow.
3. A 600 gallons tank is initially full of 2% sugar solution. Water containing 3 lb of sugar per gallon is entering at a rate of 2 gal/min, and the well-stirred mixture is allowed to flow out of the tank at a rate of 3 gal/min.
Find the amount of sugar in the tank after 30 minutes of this process.
4. A 1000 gallon holding tank that catches runoff from some
chemical process initially has 800 gallons of water with 20 ounces of
pollution dissolved in it. Polluted water flows into the tank at a rate
of 3 gal/hr and contains 5 ounces/gal of pollution in it. A well mixed
solution leaves the tank at 3 gal/hr as well. When the amount of
pollution in the holding tank reaches 500 ounces the inflow of polluted
water is cut off and fresh water will enter the tank at a decreased rate
of 2 gal/hr while the outflow is increased to 4 gal/hr. Determine the
amount of pollution in the tank after 400 hours of this process.