1) ( Applications of Differential Equations) A tank holds 100 gal of water that

Question

1) (Applications of Differential Equations)A tank holds 100 gal of water that contains 20lb of dissolved salt. A brine (salt) solution is flowing into the tank at the rate of 2 gal/min while the solution flows out of the tank at the same rate. The brine solution entering the tank has a concentration of 2 lb/gal.

a) Find an expression for the amount fo salt in the tank at any time.

b) How much salt is present after 1 hr?

c) As time increases, what happens to the salt concentration?

2) (Applications of Differential Equations)Morphine is administered to a patient intravenously at a rate of 2 mg per hour. About 25% of the morphine is metabolized and leaves the body each hour. Write a differential equation for the amount of morphine, M, in milligrams, in the body as a function of time, t, in hours.

3) (geometric series) The half life of Warfarin, an anticoagulant, is 37 hours. A patient receives an injection of 5mg of Warfarin at the same time every day for ten days. How much warfarin is in the body right after the tenth injection?

Explanation / Answer

1)So flow rate of brine=2 gal/min * 2 lb/gal = 4 lb/min

Let salt present at t =S

Salt going out=(S/100 lb/gal)*(2 gal/min) = S/50 lb/min

So dS/dt = 4 - S/50

a) dS/dt = 4 - S/50

Solving :

50 * dS/(200-S) = dt

integrating : limit : S -> 20 to S t -> 0 to 60 min

60=-50 [ ln (200-S) - ln(200-20)]

ln {(200-S)/180}=-60/50

so S=145.8 lb/gal

b) S=145.8 lb/gal

c) dS/dt > 0 if ( 4 - S/50)>0 or S<200

As time increases salt increases upto S < 200 ; After that salt decreases .

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