Question

A well mixed tank with a capacity of 1500 gals originally contains 1000 gallons of freshwater. One pipe containing 1/2 lb of salt per gallon is entering at a rate of 4 gal/min. The second pipe containing 1/3 lb of salt per gallon is entering at a rate of 6 gal/min. The mixture is allowed to flow out of the tank at a rate of 5 gal/min. Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow.

Answer 1

400 lb of salt

Explanation:

Let us assume the water flows into the rank for x minutes.

There is an initial of 1000 gallons of water in the tank and water flows in through one pipe at 4 gal/min and through another pipe at 6 gal/min. In x minute, the amount of water in the tank = 1000 + 4x + 6x = 1000 + 10x

Water flows out at 5 gal/min, therefore in x minute the amount of water in the tank = 1000 + 10x - 5x = 1000 + 5x

The tank begins to overflow when it is full (has reached 1500 gallons). Therefore:

1500 = 1000 + 5x

5x = 1500 - 1000

5x = 500

x = 100 minutes.

1/2 lb salt per gallon flows into the tank at 4 gal/min and 1/3 lb of salt is flowing in at 6 gal/min, in 100 min the amount of salt that entered the tank = 4 gal/min × 100 min × 1/2 lb/gal + 6 gal/min × 100 min × 1/3 lb/gal= 400 lb

Therefore the amount of salt is in the tank when it is about to overflow = 400 lb of salt