consider a cascade of two tanks. tank 1 contains 50 gallons of water in which 25

Question

consider a cascade of two tanks. tank 1 contains 50 gallons of water in which 25 pounds of salt are dissolved. a second tank, tank 2, contains 50 gallons of pure water. pure water is pumped into tank 1 at the rate of 3 gal/min. the mixture from tank 1 is pumped into tank 2 at the rate of 4 gal/min. the mixture from tank 2 is simultaneously pumped into tank 1 at a rate of 1 gal/min. finally the mixture is pumped out of tank 2 at the rate of 3 gal/min. let x(t) and y(t), receptively, denote the amounts of salt in tanks 1 and 2 at any time t. find the amount of salt in each tank at any time t.

Explanation / Answer

Tank 1 dx1/dt=(3gal/min)(0lbs/gal)+(1gal/min)((x2/50)lb/gal)-(4gal/min)((x1/50)lb/gal) dx1/dt=(2/25)x1+(1/50)x2 Tank 2 dx2/dt=(4gal/min)(x1/50)-(3gal/min)(x2/50)-(1gal/min)(x2/50) dx2/dt=(2/25)x1-(2/25)x2 linear system: dx1/dt=(2/25)x1+(1/50)x2 dx2/dt=(2/25)x1-(2/25)x2 Observe that the foregoing system is accompanied by the initial condition x1(0)=25, x2(0)=0

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