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15 March, 16:05

150 lb of NaCl are initially dissolved in 500 gal of a NaCl solution in a container. Water is continuously added to the container at a rate of 5 gal/min and the container content, which is kept uniform by mixing, is also withdrawn out of the container at 5 gal/min. Determine how much NaCl will be in the tank at the end of 1.5 hours.

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  1. 15 March, 16:20
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    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
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