Question

A 130 gallon tank initially contains 10 lbs salt dissolved in 60 gallons of water. Brine containing 1 lb salt per gallon begins to flow into the tank at the rate of 3 gal/min and the well-mixed solution is drawn off at the rate of 1 gal/min. How much salt is in the tank when it is about to overflow

Answer 1

80 lb of salt

Explanation:

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

There is an initial of 60 gallons of water in the tank and water flows in at 3 gal/min. In x minute, the amount of water in the tank = 60 + 3x

Water flows out at 1 gal/min, therefore in x minute the amount of water in the tank = 60 + 3x - x = 60 + 2x

The tank begins to overflow when it is full (has reached 130 gallons). Therefore:

130 = 60 + 2x

2x = 130 - 60

2x = 70

x = 35 minutes.

In 35 minutes the tank would start to overflow.

1 lb salt per gallon flows into the tank at 3 gal/min, in 35 min the amount of salt that entered the tank = 3 gal/min × 35 min × 1 lb/gal = 105 lb

1 lb salt per gallon flows out of the tank at 1 gal/min, in 35 min the amount of salt that flow out of the tank = 1 gal/min × 35 min × 1 lb/gal = 35 lb

Initially there is 10 lb of salt dissloved into 60 gallons of water.

Therefore the amount of salt is in the tank when it is about to overflow = 10 + 105 - 35 = 80 lb of salt.