query 04


 

2.5.

1.  A 3% saline solution flows at a constant rate into a 1000-gallon tank initially full of a 5% saline solution.  The solutions remain well-mixed and the flow of mixed solution out of the tank remains equal to the flow into the tank.  What constant rate of flow is necessary to dilute the solution in the tank to 3.5% in 8 hours?

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Question:  2. Solve the preceding question if the tank contains 500 gallons of 5% solution, and the goal is to achieve 1000 gallons of 3.5% solution at the end of 8 hours.  Assume that no solution is removed from the tank until it is full, and that once the tank is full, the resulting overflow is well-mixed.

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Question:  3.  Under the conditions of the preceding question, at what rate must 3% solution be pumped into the tank, and at what rate must the mixed solution be pumped from the tank, in order to achieve 1000 gallons of 3.5% solution at the end of 8 hours, with no overflow?

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Question:  4.  Under the conditions of the first problem in this section, suppose that the overflow from the first tank flows into a large second tank, where it is mixed with 3% saline solution.  At what constant rate must the 3% solution flow into that tank to achieve a 3.5% solution at the end of 8 hours?

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Question:  5. In the situation of Problem #1, suppose that solution from the first tank is pumped at a constant rate into the second, with overflow being removed, and that the process continues indefinitely.  Will the concentration in the second tank approach a limiting value as time goes on?  If so what is the limitng value?  Justify your answer.

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Now suppose that the flow from the first tank changes hour by hour, alternately remaining at a set constant rate for one hour, and dropping to half this rate for the next hour before returning to the original rate to begin the two-hour cycle all over again.  Will the concentration in the second tank approach a limiting value as time goes on?  If so what is the limiting value?  Justify your answer.

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Answer the same questions, assuming that the rate of flow into (and out of) the tank is 10 gallons / hour * ( 3 - cos(t) ), where t is clock time in hours. 

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Question:  6.  When heated to a temperature of 190 Fahrenheit a tub of soup, placed in a room at constant temperature 80 Fahrenheit, is observed to cool at an initial rate of 0.5 Fahrenheit / minute. 

If at the instant the tub is taken from the oven the room temperature begins to fall at a constant rate of 0.25 Fahrenheit / minute, what temperature function T(t) governs its temperature?

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