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Unformatted text preview: M341 H3 (S. Zhang) . 1. EP § 2.1: 11, 12, 22, 28 (Population Models) • ans: 2. Suppose that when a certain lake is stocked with fish, the birth and death rates β and δ are both inversely propor tional to √ P . Show that P ( t ) = 1 2 kt + p P 2 If P = 100 and after 6 months, there are 169 fish in the lake, how many will there be after 1 year? • ans: Let k = ( β δ ) / √ P . d dt P = k 1 √ P P Z 1 √ P dP = kdt P ( t ) = 1 2 kt + p P 2 P (0) = 100, P (6) = 169, then k = 1, and P (12) = ( t 2 + 10) 2 = 256 3. The time rate of change of an alligator population P in a swamp is proportional to the square of P . The swamp contianed a dozen alligators in 1988, two dozen in 1998. When will tehre be four dozen alligators in the swamp? What happens there after? • ans: P = kP 2 , 1 P = kt C P = 1 C kt P (0) = 12 , C = 1 12 , P = 12 1 12 kt P (10) = 24 , k = 1 240 , P = 240 20 t P = 48 , t = 15 t → 20 , P → ∞ 4. Suppose that at time t = 0, half of a logistic population of 100,000 persons have heard a certain rumor, and that the number of those who have heard it is then increasing at the rate of 1000 persons per day. How long will it take for this rumor to spread to 80% of the popuplation? • ans: Let the population unit be 1000. M = 100. P = kP ( M P ) P (0) = 1 and P (0) = 50 1 = k 50(100 50) , k = 0 . 0004 P = kP ( M P ) ⇒ P = MP P + ( M P ) e kMt When 80% reached, 80 = 100 × 50 50 + (100 50) e . 04 t t = 34 . 66....
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 Fall '08
 Zhang
 Linear Algebra, Algebra, Equations, Velocity, terminal velocity, American Alligator

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