To determine The maximum amount of fertilizer in the tank and also the time required to reach the maximum. Explanation Formula used: The differential equation describing the mixing process is given by the formula d y d t = ( chemical's arrival rate ) − y ( t ) V ( t ) ⋅ ( outflow rate ) . Calculation: Let y ( t ) be the amount of fertilizer in the tank at time t . The initial amount of water in the tank is 100 gal. So, the amount of fertilizer is 0 gal. Thus, the initial condition is y ( 0 ) = 0 . The solution containing 1 lb/gal of fertilizer enters into the tank at the rate of 1 gal/min and pumped out at a rate of 3 gal/min. Evaluate the arrival rate of the arrival of fertilizer as follows. Arrival rate = ( 1 lb gal ) ( 1 gal min ) = 1 ( lb min ) Thus, the arrival rate of the concentrate is 1 ( lb min ) . Evaluate the rate of change of the amount of fertilizer in tank is evaluated as follows. ( Rate of change of amount in container ) = ( rate at which chemical arrives ) − ( rate at which chemical departs ) = 1 gal − 3 gal = − 2 gal Thus, 2 gal of fertilizer is pumped out from the tank for every 1 min. At time t the volume of the tank is given by ( 100 − 2 t ) gal . Evaluate the differential equation describing the mixing process as follows. d y d t = ( chemical's arrival rate ) − y ( t ) V ( t ) ⋅ ( outflow rate ) d y d t = 1 − y 100 − 2 t ( 3 ) d y d t + 3 100 − 2 t y = 1 Thus, the initial value problem is d y d t + 3 100 − 2 t y = 1 , y ( 0 ) = 0 . Compare the above equation with the standard equation d y d t + P ( x ) y = Q ( x ) . Here, P ( t ) = 3 100 − 2 t and Q ( t ) = 1 . Evaluate the integrating factor as follows. v ( t ) = e ∫ P ( t ) d t = e ∫ 3 100 − 2 t d t = e − 3 2 ln ( 100 − 2 t ) = e ln ( 100 − 2 t ) − 3 2 = ( 100 − 2 t ) − 3 2 Evaluate the solution as follows. y = 1 v ( t ) ∫ v ( t ) Q ( t ) d t = 1 ( 100 − 2 t ) − 3 2 ∫ ( 100 − 2 t ) − 3 2 ( 1 ) d t = 1 ( 100 − 2 t ) − 3 2 ( − 2 ( 100 − 2 t ) − 1 2 − 2 + C ) = ( 100 − 2 t ) 3 2 ( 100 − 2 t ) − 1 2 + C ( 100 − 2 t ) 3 2 = ( 100 − 2 t ) + C ( 100 − 2 t ) 3 2 The initial amount of fertilizer in the tank is 0 gal

14th Edition

ISBN: 9780134654874

Author: WEIR

Publisher: VST

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Chapter 9.3, Problem 15E

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The maximum amount of fertilizer in the tank and also the time required to reach the maximum.

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Chapter 9.3, Problem 14E

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