A tank originally contains 120 liters of water with 20 grams of salt in solution

Question

A tank originally contains 120 liters of water with 20 grams of salt in solution. Beginning at t=0, water containing 0.6 grams of salt per liter flows into the tank at a rate of 2 liters per minute and the uniform mixture drains from the tank at a rate of 2 liters per minute. Letting t be time in minutes and Q be the amount of salt in the tank at time t measured in grams, formulate an initial value problem modeling the amount of salt in the tank at any time.

Find the solution of the initial value problem

Explanation / Answer

rate of salt inflow =0.6*2 =1.2 gm/min

rate of salt outflow =2*Q(t)/120 =Q(t)/60 gm/min

rate of change of salt =rate of saltinflow -ratte of salt outflow

dQ/dt=1.2 -Q(t)/60

Q(0)=20

dQ/dt=1.2 -Q(t)/60

dQ+Q(t)/60 dt=1.2 dt

integrating factor =e 1/60 dt

integrating factor =e1/60 t

multiiply with integrating factor on both sides

dQ e1/60 t+e1/60 t(1/60)dtQ(t)=1.2e1/60 t dt

(Q(t)e1/60 t)'==1.2e1/60 t dt

integrate on both sides

(Q(t)e1/60 t)'= 1.2e1/60 t dt

(Q(t)e1/60 t)= 60*1.2e1/60 t +C

(Q(t)e1/60 t)= 72e1/60 t +C

Q(t)= 72 +Ce-1/60 t

Q(0)=20

72 +Ce-1/60 *0=20

C=20-72

C=-52

SOLUTION IS Q(t)= 72 -52e-1/60 t

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.