Question

1. Find the differential equation and the corresponding initial condition modeling the following situation. However, do not solve the resulting initial value problem! A 500-gallon tank initially contains 200 gallons of sugar water that was made by dis- solving 8 pounds of sugar in the water. Sugar water with a concentration of 0.30 pounds of sugar per gallon is pumped into the tank at the rate of 10 gallons per minute, while the sugar water in the tank is pumped out at the rate of 8 gallons per minute. What is the differential equation whose solution would represent the concentration of the sugar in the tank at time t minutes (up to the moment that the tank begins overflowing)? What is the corresponding initial condition?

Answer 1

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Solution:-

sugar let fest. The amount of hole in the tank at time & Vin(t) = The rate at which sugas water is flowing into Tout (4) a flow-out rate Cm (t) = Concentration of sugar in the in-flowing mistere - at time t o Coul (U= concentration of sugar in the out-flowing misture at time t Sin(t) = 10 rout (4)= 8. Cm (4) 2 0.30 care constants) b sx = salt in a fald out » (Cindin - Cont Yout) at ( 3 - 8 Cout) 91 s om = 3- Skut) (3-8 Cout) Now Cout (a) . Cout (4) elt : Where V(t) is the wlume of fluid VG+) in the tank as time t. Ance, the initial vluste is 200 gallon & a net total of cron-dout) = (10.8) = 2 gallon is added every minute de VG) 200 +27 So, we're di Herendial equatia do 3 2 00+27 at t=0, x= 8 de s dx = 3 - 4 latz = 3. 8 l Sitial Gondon Condition A = dx = 296 +37) la + t

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