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