4. A tank containing sugar water is set up so that a sugar-mixture containing 3

Question

4. A tank containing sugar water is set up so that a sugar-mixture containing 3 pound of sugar per gallon is flowing in at a rate of 7 gallons per minute. The sugar mixture is well mixed, and leaves the tank at a rate of 2 gallons per minute. Initially, the tank contained one pound of sugar. Set up, but do not solve a differential equation, with initial condition, that corresponds to this scenario. Carefully describe where each term in your differential equation came from. (a) For your equation from question 4, where is the fact that the tank is gaining volume represented in your equation? (b) For your equation from question 4, where is the fact that the tank is well mixed represented in your equation?

Explanation / Answer

4.

x(t)= a function which tells the amount of sugar in the tank at time t

Inflow Information :

ri= 7 gal/min

ci= 3 lb/gal

outflow Information:

ro= 2 gal/min

co=?

Volume:

Initial volume = V0

Formula: V(t)= V0 +( ri-r0)t= V0+(7-2)t = V0+5t

Concentration:

co=x(t)/V(t) = x(t)/(V0 +5t)

Initial condition:

x(0)= 1

differential equation:

dx/dt = rici - roco

dx/dt = 21 - 2x/(V0+5t)

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