A tank initially contains s0 lb of salt dissolved in 100 gal of water, where s0 is some positive number. Starting at t = 0, water containing 0.5 lb of salt per gallon enters the tank at a rate of 2 gal/min, and the well-stirred solution leaves the tank at the same rate. Letting c(t) be the concentration of salt at time t, show that the limiting concentration–i.e., limt→∞ c(t)–is 0.5 lb/gal. (1) Set up an initial value problem using the situation described above. It may help to draw a diagram. (2) Solve the differential equation, and use the initial value to solve the initial value problem, and use this function to write an explicit formula for c(t). (3) Show that limt→∞ c(t) = 0.5. (4) Discuss what this limit implies about the importance of the unknown quantity s0.
A tank initially contains s0 lb of salt dissolved in 100 gal of water, where s0 is some positive number. Starting at t = 0, water containing 0.5 lb of salt per gallon enters the tank at a rate of 2 gal/min, and the well-stirred solution leaves the tank at the same rate. Letting c(t) be the concentration of salt at time t, show that the limiting concentration–i.e., limt→∞ c(t)–is 0.5 lb/gal.
(1) Set up an initial value problem using the situation described above. It may help to draw a diagram.
(2) Solve the differential equation, and use the initial value to solve the initial value problem, and use this function to write an explicit formula for c(t).
(3) Show that limt→∞ c(t) = 0.5.
(4) Discuss what this limit implies about the importance of the unknown quantity s0.
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