Answer:
After 90 minutes there will be 60.5 lb NaCl in the tank
Explanation:
Initially, you have 150 lb/500 gal = 0.300 lb/gal NaCl
After 1 minute, the volume will be the same but you will have less NaCl in the solution.
After the first minute, the mass of NaCl you will lose will be:
0.3 lb/gal · 5 gal = 1.5 lb
Then, the concentration of the solution will be:
(150 lb - 1.5 lb)/ 500 gal = 0.297 lb/gal
After the second minute:
mass of NaCl lost = 0.297 lb/gal · 5 gal = 1.485 lb
concentration of the solution = (148.5 lb - 1.485 lb) / 500 gal = 0.29403 lb/gal
After the third minute:
mass of NaCl lost = 0.294 lb/gal · 5 gal = 1.47015 lb
concentration of the solution = 147.015 lb - 1.47015 lb / 500 gal = 0.2910897 lb/gal
Notice that the solution is being diluted by 1% every minute. Then, the concentration of the solution after every minute will be 99% of the previous solution. For example, after the first minute, the concentration of the solution will be 99% of the original solution (99% of 0.3 lb/gal = 0.297 lb/gal). Mathematically, we can express it as follows:
0.3 lb/gal · 0.99 = 0.297 lb/gal
After the second minute:
0.297 lb/gal · 0.99 = 0.29403 lb/gal
Since 0.297 lb/gal = 0.3 lb/gal · 0.99
we can write for the second minute:
0.3 lb/gal · 0.99 · 0.99 = 0.29403 lb/gal
And for the third minute:
0.3 lb/gal · 0.99 · 0.99 · 0.99 = 0.2910897 lb/gal
And for the nth minute:
0.3 lb/gal · 0.99ⁿ = concentration of the solution after n minutes
Now, we can find the concentration of the solution after 90 min:
0.300 lb/gal · 0.99^90 = 0.121 lb/gal
The mass of NaCl after 90 minutes will be:
0.121 lb/gal · 500 gal = 60.5 lb
