Question

A tank originally contains 100 gal of fresh water. Then water containing 12 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of 4 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. (Round your answer to two decimal places.)

Answer 1

Answer:

The amount of salt in the tank will be 1.60 lb

Explanation:

After the first minute of the process, the tank will have a concentration of 0.24 lb/ gal:

amount of salt added / volume of the tank = concentration in the tank

2 gal · 12 lb/gal /100 gal = 0.24 lb/gal

After the minute 2, 24 lb of salt (2 gals · 12 lb/gal) will be incorporated to the 100 gal and the final concentration of the tank will be:

(24 lb + 24 lb) /100 gal = 0.48 lb /gal

Then, every minute the concentration of the tank will increase by 0.24 lb/gal

The concentration of the tank after 10 min will be:

Concentration of salt in the tank after 10 min = 0.24 lb/gal · 10 = 2.40 lb/gal

Then, freshwater is added at a rate of 4 gal/min. The solution is being diluted so that the concentration of the resulting solution is 96% of the previous solution. After minute 1, the solution will have a concentration that is 96% of the original solution:

Concentration of the solution after minute 1 = 2.40 lb/gal · 0.96 = 2.304 lb/gal

After minute 2:

2.304 lb/ gal · 0.96 = 2.21184 lb/gal

Since 2.304 lb/gal = 2.40 lb/gal · 0.96

we can rewrite the expression for minute 2:

2.40 lb/gal · 0.96 · 0.96 = 2.21184 lb/gal

And for minute 3:

2.40 lb/gal · 0.96 · 0.96 · 0.96= concentration after 3 minutes

And for minute n

2.40 lb/gal · 0.96ⁿ = concentration after n minutes

Then, after 10 minutes of this process:

2.40 lb/gal · 0.96¹⁰ = 1.60 lb/gal

The amount of salt in the tank will be:

1.60 lb/gal · 100 gal = 160 lb salt