Math Question

14. Two tanks, Tank I and Tank II, are filled with \( V \) gal of pure water. A solution containing \( a \mathrm{lb} \) of salt per gallon of water is poured into Tank I at a rate of \( b \) gal per minute. The solution leaves Tank \( \mathrm{I} \) at a rate of \( b \mathrm{gal} / \mathrm{min} \) and enters Tank II at the same rate \( (b \mathrm{gal} / \mathrm{min}) \). A drain is adjusted on Tank II and solution leaves Tank II at a rate of \( b \mathrm{gal} / \mathrm{min} \). This keeps the volume of solution constant in both tanks ( \( V \) gal). Show that the amount of salt solution in Tank II, as a function of time \( t \), is given by \( a V-a b t e^{-(b / V) t}-a V e^{-(b / V) t} . \)



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