Please answer question 1. Note the picture below the question is what the answer

Question

Please answer question 1. Note the picture below the question is what the answer should be! Hence, Q_1=2-1/10 Q1+1/25Q2 and Q_2=6+3/50Q1-1/20Q2. Please fully show how to get these two answers!!!

1. Tanks Ti and T2 contain 50 gallons and 100 gallons of salt solutions, respectively. A solution with 2 pounds of salt per gallon is pumped into Ti from an external source at 1 gal/min, and a solution with 3 pounds of salt per gallon is pumped into T2 from an external source at 2 gal/min. The solution from Ti is pumped into T2 at 3 gal/min, and the solution from T2 is pumped into Ti at gal/min. Ti is drained at 2 gal/min and T2 is drained at 1 gal/min. Let Qit) and Q2() be the number of pounds of salt in Ti and Tz. respectively, at time 1>0. Derive a system of differential equations for Qi and Q2. Assume that both mixtures are well stirred. 10 25 10.1.1 (p. 515) SO

Explanation / Answer

Let us look at this step by step-

Amount of liquid in T1 is 50 gallon, T2 is 100 gallon.

First Let us convince ourself that the amount of liquid(not concentration ) remains the same in each tanks.

T1

1 gallon in from external

3 gallon out to T2

4 gallon in from T2

2 gallon out to external.

1-2-3+4=0

T2

2 gallon in from external

4 gallon out to T1

3 gallon in from T1

1 gallon out to external

2-4+3-1=0

Therefore the concentration of liquid in T1 and T2 will be Q1/50 and Q2/100 all the time.

Now lets check in T1

step by step

1)2 pound of salt per gallon is pumped in at a rate of 1 gallon per minute:

Q1'=2......

2)Solution from T1 is pumped into T2 at a rate of 3 gallon per minute.

Q1'=2 - (Q1/50)*3........                (Remember why we used Q1/50 ?)

3)Solution from T2 is pumped in T1 at 4 gallon per minute

Q1'=2-(Q1/50)*3+(Q2/100)*4.........

4)Solution from T1 is drained at a rate of 2 gallon per minute.

Q1'=2-(Q1/50)*3+(Q2/100)*4-(Q1/50)*2

Q1'=2-(Q1/10)+(Q2/25)

Similarly we can proceed for Tank2:-

1)A solution with 3 pounds of salt per Gallon is pumped in at a rate of 2 gallon per minute.

Q2'=3*2..................

2)Solution from T1 is pumped into T2 at a rate of 3 gallon per minute.

Q2'=6+(Q1/50)*3..............

3)Solution is pumped from T2 to T1 at a rate of 4 gallon per minute.

Q2'=6+(Q1/50)*3-(Q2/100)*4..........

4)The tank is drained at a rate of 1 gallon per minute.

Q2'=6+(Q1/50)*3-(Q2/100)*4-(Q2/100)

Q2'=6+(3*Q1/50)-Q2/20

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