Question

A tank originally contains 5lb of salt dissolved in 200gal of water. Starting at time t=0, a salt solution containing 0.10lb of salt per gallon is to be pumped into the tank at a constant rate and the well-stirred mixture is to flow out of the tank at the same rate. (a) The pumping is to be done so that the tank contains 15 lb of salt after 20 min of pumping. At what rate must the pumping occur in order to achieve this objective? (b) Suppose the objective is to have 25lb of salt in the tank after 20 min. Is it possible to achieve this objective? Explain.

Answer 1

The required pumping rate to achieve a salt concentration of 15lb after 20 minutes is 0.5lb/min. However, if the aim were to achieve 25lb of salt in the tank, it would be impossible given the constraints of the problem - the inflow rate needed exceeds the capacity of the tank.

Step-by-step explanation:

For part (a) of the problem, we begin by figuring out how much additional salt needs to be added to the tank. Given that the goal is to have a final amount of 15lb and we initially have 5lb, this means we need to add an extra 10lb of salt. Hence, if this addition happens over 20 mins, the pumping rate required can be calculated by dividing the extra salt by the given time. Therefore, it is 10lb/20min = 0.5lb/min. This is how fast the salt solution must be pumped into the tank to achieve the desired salt concentration.

For part (b), if the objective is to have 25lb of salt in the tank after 20 min, it would imply we need to add an extra 20lb of salt (25lb-5lb). Given the highest concentration of incoming salt solution is 0.10lb per gallon, we would have to pump in 200 gallons of solution to get this amount of salt. This would be 200 gallons in 20 minutes, or a rate of 10gal/min. Unfortunately, the system cannot both add 200 gallons in 20 min and keep the tank volume constant at 200 gallons, as the same amount of solution is also flowing out. Therefore, it is not possible to achieve this objective.

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