Rate In > Rate Out Initially, a large tank with a capacity of 100 gallons contai

Question

Rate In > Rate Out Initially, a large tank with a capacity of 100 gallons contains 50 gallons of pure water. A salt solution with a concentration of 0.1 lb/gal flows into the tank at a rate of 4 gal/min. The mixture is kept well-stirred and flows out of the tank at the rate of 2 gal/min. (a) Find the initial-value problem that describes the amount of salt in the tank. (b) Find the amount of salt in the tank until the tank overflows. (c) Find the concentration of salt in the tank until the tank overflows. (d) What is the initial-value problem that describes the amount of salt in the tank after the tank overflows?

Explanation / Answer

Given: Initial volume of tank = 500 gallons volume flow rate in = 20 gallons/minute volume flow rate out = 10 gallons/minute Let m(t) = amount of particulate in the tank at any time t m'(t) = rate of change in amount of particulate in the tank c(t) = concentration of particulate in the tank at any time t (lb/gallon) m'(t) = rate of particulate added - rate of particulate removed rate of addition = 0

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