The population of a certain breed of rabbit in a wildlife sanctuary t months aft

Question

The population of a certain breed of rabbit in a wildlife sanctuary t months after they've been introduced to a new habitat is given by R(t) = 200- 100e^-0.06t cos(0.2t) rabbits. (a) How many rabbits were present when they were introduced to their new habitat? (b) At what rate was the rabbit population changing initially? (c) How many rabbits were in the sanctuary, and at what rate was the rabbit population changing, 20 months after they were introduced? (d) Use limits to study the long-term behavior of R(t). What does this imply about the rabbit population? (e) Support the above graphically by plotting R(t) for t in [0, 100]. Sketch the graph, labeled appropriately.

Explanation / Answer

given

R(t) = 200-100e^(-0.06t)cos(0.2t)

a) to calculate the rabbits initially t= 0

=> R(0) = 200 - 100e^(0) cos(0)

= R(0) = 200-100 = 100rabbits

b) rate at which the rabbit poplutaion changed initially was

R'(t) = 0-[100(-0.06)e^(-0.06t)cos(0.2t) + 100e^(-0.06t)(0.2)(-sin(0.2t))]

=> R'(t) = 6e^(-0.06t) .cos(0.2t) + 20e^(-0.06t).sin(0.2t)

to find the intial change t = 0

=> R'(0) = 6.e^(0) .cos(0) + 20(e^(0) .sin(0) = 6

=> the rate of change of rabbits initially are 6

c) no of rabbits when t = 20

R(20) = 200-100.e^(-0.06*20).cos(0.2*20) = 170

rate of change of rabbits when t = 20 is

R'(t) = 6e^(-0.06*20) .cos(0.2*20t) + 20e^(-0.06*20).sin(0.2*20)

= 2.22 = 2

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