Explanation Given: The below figure shows all given data. Figure (1) Formula used: The rate of change of volume for Δ t → 0 is given by, d V d t = r 1 − r 2 …… ( 1 ) Here, the flow rate of the chemical into the tank is r 1 , the rate at which the mixture flow out of the tank is r 2 and rate of change of the volume is d V d t . The rate of change of amount of chemical is given by, d A d t = c 1 r 1 − c 2 r 2 …… ( 2 ) Here, the concentration of salt flowing into the solution is c 1 and the rate of change of amount of chemical is d A d t . The final volume of the tank is given by, V ( t ) = ( r 2 − r 1 ) t + V ( 0 ) …… ( 3 ) Here, the initial volume is V ( 0 ) and the time is t . The concentration of chemical in the tank at any time is, c 2 = A ( t ) V ( t ) …… ( 4 ) Here, the amount of chemical is A ( t ) . Calculation: The expression for the rate of change of volume of the chemical in the tank 1 using equation ( 1 ) is, d V A d t = r 1 − r 2 Integrate the above equation. V A ( t ) = ( r 1 − r 2 ) t + V 1 …… ( 5 ) Here, the initial volume of the tank A is A 1 . The expression for the rate of change of amount of the solution in the tank using equation ( 2 ) is, d A 1 d t = c 1 r 1 − c 2 r 2 Here, the area of the solution in the tank is A 1 . Substitute ( A V ) for c 2 in the above equation using equation ( 5 ) . d A 1 d t = c 1 r 1 − ( A V ) r 2 …… ( 6 ) Substitute ( r 1 − r 2 ) t + V 1 for V in equation ( 6 ) to get the rate of change of amount of chemical in the tank A. d A 1 d t = c 1 r 1 − ( A 1 ( r 1 − r 2 ) t + V 1 ) r 2 d A 1 d t + ( A 1 ( r 1 − r 2 ) t + V 1 ) r 2 = c 1 r 1 d A 1 d t + ( r 2 ( r 1 − r 2 ) t + V 1 ) A 1 = c 1 r 1 …… ( 7 ) The expression for the rate of change of volume of the chemical in the tank 2 using equation ( 1 ) is, d V B d t = r 2 − r 3 Here, the flow rate of the chemical into the tank is r 2 , the rate at which the mixture flow out of the tank is r 3 Integrate the above equation. V B ( t ) = ( r 2 − r 3 ) t + V 2 …… ( 8 ) Here, the constant is V 2 . The expression for the rate of change of amount of the solution in the tank 2 using equation ( 2 ) is, d A 2 d t = c 2 r 2 − c 3 r 3 Here, the area of the solution in the tank is A 2 To determine (b) To find: The amount of the chemical in the second tank.

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Question

Chapter 1.7, Problem 8P

To determine

(a)

To find:

Whether the amount of chemical in the tank at time t is A1(t)=dA1dt+r2(r1−r2)t+V1A1=c1r1 and A2(t)= dA2dt+r3(r2−r3)t+V2A2=r2A1(r1−r2)t+V1 or not.

To determine

(b)

To find:

The amount of the chemical in the second tank.

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Chapter 1.7, Problem 7P

Chapter 1.7, Problem 9P

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