hi there, Please explain how you reach the answer, Two tanks (each can hold 100

Question

hi there,

Please explain how you reach the answer,

Two tanks (each can hold 100 gallons) are interconnected. Tank 1 initially contains 30 gallons of water and 2 lbs of salt. Tank 2 initially contains 40 gallons of water and 5 lbs of salt. Water containing 1 lb of salt per gallon flows into Tank 1 at a rate of 2 gallons per minute. Solution flows from Tank 1 into Tank 2 at a rate of 2 gallons per minute. Solution also flows from Tank 2 into Tank 1 at a rate of 1 gallon per minute. Solution drains out of Tank 2 at a rate of 1 gallon per minute. Let Q1 (t) = lbs of salt in Tank 1 at time t and Q2(t) = lbs of salt in Tank 2at time t. Set up the system of 2 differential equations, with initial conditions that describe the amount of salt in these tanks. State how long your system is valid for.

Explanation / Answer

Initial Volume of Tank 1 = 100 gallons
Initial Volume of Tank 2 = 100 gallons

First, you have to note the inflow and outflow of water for each tank:
Tank 1: In= (2+1)gallons/min ??Out= 2 gallons/min
Tank 2: In= 2gallons/min??? Out= 2 gallons/min ---->(1+1)

Now let Q1(t) = amount of salt, in grams, in Tank 1 as function of time per min
?and?Q2(t) = amount of salt, in grams, in Tank 2 as function of time per min
where at time = 0 --> Q1(0) = 2 lbsof salt
? ? ? ?? ? ? Q2(0) = 5 lbs grams of salt

Thus the derivative is:
? Q1'(t) = the rate of change with respect to time of the amount of salt in Tank 1
? Q2'(t) = the rate of change with respect to time of the amount of salt in Tank 2

Q1'(t) = (Incoming flow in Tank 1)(concentration fo salt in flow)
???+ (Incoming flow from Tank 2)(Concentration of salt in Tank 2)
???- (Outgoing flow from Tank 1)(concentration in Tank 1)

Q1'(t) = (Incoming flow)(concentration of salt in flow)
???+ (Incoming flow from Tank2)[p(t)/(Volume of water in Tank 2)]
???- (Outgoing flow from Tank 1)[q(t)/(Volume of water in Tank 1)]

Q1'(t) = (2 gallon/min)(1 lb/gallon) + (1 gallon/min)[Q1(t)/100] - (2gallon/min)Q2(t)/100L]

Q1'(t) = 2 + (1/100)Q1(t) - (1/50)Q2(t)


Q2'(t) = (Incoming flow)(Concentration of salt in flow)
???+ (incoming flow from Tank 1)[p(t)/(Volume of water in Tank 1)]
???- (Outgoing flow from Tank2)[q(t)/(Volume of water in Tank 2)]

similarly put the values !!!

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