midterm_1_review (dragged) 2

midterm_1_review (dragged) 2 - MA 36600 MIDTERM#1 REVIEW 3...

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MA 36600 MIDTERM #1 REVIEW 3 § 2.3: Modeling with First Order Equations. Mixing Problems: Say that we have a tank containing a volume V gal of a certain liquid. A concentration c lbs/gal of a certain substance flows in at a constant rate r gal/min, and a faucet is opened at the bottom of the tank so that the total amount of liquid in the tank remains constant. Let Q = Q ( t ) denote the amount in lbs of this substance at time t , where Q (0) = Q 0 . Concentration of Substance Rate of Flow Flowing In c lbs/gal r gal/min Flowing Out Q ( t ) /V lbs/gal r gal/min The rate of change of the quantity of the substance in the solution is the di ff erence of the rate which flows in and the rate which flows out. This gives the following initial value problem: dQ dt + r V Q = c r, Q (0) = Q 0 . Interest Rate Problems: Say that we have an account where money accrues interest compounded continuously at an annual rate r , and that deposits are also made at a constant rate k per year. Let S = S ( t ) denote the amount of money in the account at t
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