Section 1.5. Solving the Real Applications Using Differential Equations
Objective:
1. Set up a differential equation for the real word application
2. Solve the application problems
Fundamental question: At time [latex]t=0,[/latex] a tank contains [latex]A[/latex] gal of
salt solution with concentration [latex]a[/latex] lb/gal. Assume that solution
containing [latex]b[/latex] lb/gal salt is entering tank at rate of [latex]r_{in}[/latex]
gal/min, and leaves at rate at [latex]r_{out}[/latex]. Assuming the solution is
well mixed before leaving the tank.
(a) Set up IVP that describes this salt solution flow process.
(b) Find amount of salt [latex]Q(t)[/latex] in tank at any given time [latex]t[/latex].
(c) Find limiting amount [latex]Q_{L}[/latex] of salt [latex]Q(t)[/latex] in tank after a very long time.
Example 1: A mixing tank initially contains [latex]100[/latex] liters of fresh water. Solution containing [latex]10[/latex] grams/liter of carbon dioxide flows into the tank at the rate of 2 liters/min. The well-stirred mixture flows out of the tank at the same rate.
(a) Set up an IVP modeling the process.
(b) Solve the IVP obtained.
Exercise 1: A mixing tank initially contains [latex]150[/latex] liters of fresh water. Solution containing [latex]5[/latex] grams/liter of carbon dioxide flows into the tank at the rate of 3 liters/min. The well-stirred mixture flows out of the tank at the same rate.
(a) Set up an IVP modeling the process.
(b) Solve the IVP obtained.
Example 2: A mixing tank initially contains [latex]40[/latex] liters of fresh water. Solution containing [latex]5[/latex] grams/liter of carbon dioxide flows into the tank at the rate of 2 liters/min. The well-stirred mixture flows out of the tank at the rate of [latex]4[/latex] liters/min.
(a) Set up an IVP modeling the process.
(b) Solve the IVP obtained.
Exercise 2: A mixing tank initially contains [latex]150[/latex] liters of fresh water. Solution containing [latex]5[/latex] grams/liter of carbon dioxide flows into the tank at the rate of 3 liters/min. The well-stirred mixture flows out of the tank at the rate of [latex]7[/latex] liters/min.
(a) Set up an IVP modeling the process.
(b) Solve the IVP obtained.
Example 3: A mixing tank initially contains [latex]40[/latex] liters of solution with concentration [latex]1[/latex] gram/liter. Fresh water flows into the tank at the rate of 2 liters/min. The well-stirred mixture flows out of the tank at the same rate.
(a) Set up an IVP modeling the process.
(b) Solve the IVP obtained.
(c) Find the time when the tank reaches [latex]1\%[/latex] of its original value.
Exercise 3: A mixing tank initially contains [latex]40[/latex] liters of solution with concentration [latex]1[/latex] gram/liter. Fresh water flows into the tank at the rate of 2 liters/min. The well-stirred mixture flows out of the tank at the same rate.
(a) Set up an IVP modeling the process.
(b) Solve the IVP obtained.
(c) Find the time when the tank reaches [latex]20\%[/latex] of its original value.
Group Work:
1. Consider a pond that initially contains 10 million gallons of fresh water. Water containing toxic waste flows into the pond at the rate of 5 million gal/year, and exits at same rate. The concentration [latex]c(t)[/latex] of toxic waste in the incoming water varies periodically with time: [latex]c(t)=1+\text{sin}(t)[/latex] grams/gal. Determine [latex]Q(t)[/latex] of toxic waste in the pond at any time [latex]t[/latex].
2. A tank with capacity [latex]500[/latex] gal initially contains [latex]400[/latex] gal of water with [latex]40[/latex] lb of salt in solution. Water containing [latex]2+\text{sin (20t)[/latex] lb of salt per gallon entering at a rate of [latex]3[/latex] gal/min, and mixture is allowed to flow out at a rate of 1 gal/min. Let [latex]Q(t)[/latex] be the amount of salt in the tank at time [latex]t[/latex]. Set up an IVP to find [latex]Q(t)[/latex]. Do not solve the [latex]Q(t)[/latex].
3. A tank with a capacity of 400 liters originally contains 200 liters hydrogen peroxide solution with concentration of 3 grams of [latex]H_{2}O_{2}[/latex] per liter water. Additional hydrogen peroxide solution of concentration 6 grams/liter flows into the tank at a rate of 5 liters/min. The well mixed solution leaves the tank at a rate of 3 liters/min. Find [latex]H(t)[/latex], the among of [latex]H_{2}O_{2}[/latex] at any time [latex]t[/latex].