Water containing concentration c lb/gal of salt is pored into a tank while water
ID: 1890704 • Letter: W
Question
Water containing concentration c lb/gal of salt is pored into a tank while water is drained out at a faster rate. For simplicity assume that the rate in is 1 gal/min and the rate out is 2 gal/min, and the tank initially contains 100 gallons of water and x0 lbs of salt.a)Show that if concentration c of the water entering the tank is the same as the initial concentration of salt (= x0 /100), then the amount of salt x(t) is a straight line
b) Show what the solution x(t) looks if c > x0 /100 graph
c) Show what the solution x(t) looks if c < x0 /100 graph
Explanation / Answer
A) (accumulation, lb/min) = (rate in, lb/min) - (rate out, lb/min) dQ/dt = 2*3 - (Q/300)*3 = 6 - Q/100 dQ/(6-Q/100) = dt -100*ln(6-Q/100) = t + D ln(6-Q/100) = -t/100 + C 6-Q/100 = B*exp(-t/100) 600-Q=A*exp(-t/100) Q = 600-A*exp(-t/100) initial condition ==> t=0, Q=50 ==> A=550 Q = 600-550*exp(-t/100) (lb) B) C = Q/300 C = (600-550*exp(-t/100))/300 C = 2-11/6*exp(-t/100) (lb/gal) C) steady-state ==> dQ/dt=0 dQ/dt = 6 - Q/100 0 = 6 - Q/100 Q = 600 lb D) steady-state ==> Q=600 (from part C) C = Q/300 C = 600/300 C = 2 lb/gal
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