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13 November, 05:15

Initially, 50 pounds of salt are dissolved in a large tank holding 300 gallons of water. A brine solution with a concentration of 2 pounds per gallon is pumped into the tank at a rate of 3 gallons per minute, and the well-stirred solution is then pumped out at the same rate. Write a differential equation to model this problem, and use it to determine a function describing the amount of salt in the tank at any given time. How much salt will be in the tank if this process is allowed to continue for an infinite amount of time

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  1. 13 November, 07:00
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    dA / dt = 6 - A (t) / 100

    600

    Step-by-step explanation:

    We have to:

    dA / dt = In - Out

    Input = input salt concentration * input rate of brine

    replacing

    Inlet = 2 lb / gal * 3 gal / min = 6 lb / min

    there is no build up because the inlet flow equals the outlet, so there are 300 gallons in the tank at the start

    Output: salt concentration at the output * output rate of brine

    replacing

    Output: A (t) / 300 * 3 gal / min = A (t) / 100

    Thus:

    dA / dt = 6 - A (t) / 100

    this would be the differential equation.

    now, when time tends to infinity, we would be left with:

    100 * dA / dt = 600 - A (t)

    When t tends to infinity, A (t) tends to 0, therefore, the amount of salt would be 600.
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