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16. A tank initially holds V0 liters of brine (salty water) that contains a kilograms of salt. Another brine solution containing b kilograms of salt per liter is poured into the tank at the rate of e liters per minute while, simultaneously, the well-stirred solution leaves the tank at the rate of f liters per minute. The differential equation for the amount of salt Q in the tank at any time t is:
. . . . .\(\displaystyle \dfrac{dQ}{dt}\, +\, \dfrac{f}{V_0\, +\, (e\, -\, f)t}\, Q\, =\, be\)
a. A tank initially holds 100 liters of a brine solution containing 20 kilograms of salt. At t = 0, fresh water (with no salt in it) is poured into the tank at the rate of 5 liters per minute, while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time.
b. A tank initially holds 100 liters of a brine solution containing 1 kilogram of salt. At t = 0, another brine solution containing 1 kilogram of salt per liter is poured into the tank at the rate of 3 liters per minute, while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time.
c. A tank with the capacity of 50 liters contains 10 liters of fresh water (with no salt in it). At t = 0, a brine solution containing 1 kilogram of salt per liter is poured into the tank at the rate of 4 liters per minute, while the well-stirred mixture leaves the tank at the rate of 2 liters per minute. Find the amount of time required for overflow to occur. (overflow: mixture present in the tank = initial capacity, 50 liters, of the tank)
16. A tank initially holds V0 liters of brine (salty water) that contains a kilograms of salt. Another brine solution containing b kilograms of salt per liter is poured into the tank at the rate of e liters per minute while, simultaneously, the well-stirred solution leaves the tank at the rate of f liters per minute. The differential equation for the amount of salt Q in the tank at any time t is:
. . . . .\(\displaystyle \dfrac{dQ}{dt}\, +\, \dfrac{f}{V_0\, +\, (e\, -\, f)t}\, Q\, =\, be\)
a. A tank initially holds 100 liters of a brine solution containing 20 kilograms of salt. At t = 0, fresh water (with no salt in it) is poured into the tank at the rate of 5 liters per minute, while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time.
b. A tank initially holds 100 liters of a brine solution containing 1 kilogram of salt. At t = 0, another brine solution containing 1 kilogram of salt per liter is poured into the tank at the rate of 3 liters per minute, while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time.
c. A tank with the capacity of 50 liters contains 10 liters of fresh water (with no salt in it). At t = 0, a brine solution containing 1 kilogram of salt per liter is poured into the tank at the rate of 4 liters per minute, while the well-stirred mixture leaves the tank at the rate of 2 liters per minute. Find the amount of time required for overflow to occur. (overflow: mixture present in the tank = initial capacity, 50 liters, of the tank)
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