A particular region has a rabbit population of 1600. Two foxes are introduced to control the population of rabbits. Following this, the number of rabbits decreases according to the formula R(t) = 1700 – Aekt. where A and k are constants, and R(t) is the number of rabbits in the region t years after the introduction of the foxes. (a) Given that the population of rabbits drops by one quarter after 5 years, find the values of A and k. (b) Following this model, how long will it take for the rabbits to become extinct? Give your answer to two decimal places. (c) Let F(t) be the number of foxes in the region t years after their introduction. If dF = 0.7F(t), dt find the time at which the rate of decrease of the rabbit population is equal to the rate of increase of the fox population, correct to two decimal places. dF and dt dR Hint. Note that represent the rates of change of the rabbit and fox dt populations respectively.

A particular region has a rabbit population of 1600. Two foxes are introduced to control the population of rabbits. Following this, the number of rabbits decreases according to the formula R(t) = 1700 – Aekt. where A and k are constants, and R(t) is the number of rabbits in the region t years after the introduction of the foxes. (a) Given that the population of rabbits drops by one quarter after 5 years, find the values of A and k. (b) Following this model, how long will it take for the rabbits to become extinct? Give your answer to two decimal places. (c) Let F(t) be the number of foxes in the region t years after their introduction. If dF = 0.7F(t), dt find the time at which the rate of decrease of the rabbit population is equal to the rate of increase of the fox population, correct to two decimal places. dF and dt dR Hint. Note that represent the rates of change of the rabbit and fox dt populations respectively.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A particular region has a rabbit population of 1600. Two foxes are introduced to
control the population of rabbits. Following this, the number of rabbits decreases
according to the formula
R(t) = 1700 – Aekt.
-
where A and k are constants, and R(t) is the number of rabbits in the region t years
after the introduction of the foxes.
(a) Given that the population of rabbits drops by one quarter after 5 years, find
the values of A and k.
(b) Following this model, how long will it take for the rabbits to become extinct?
Give your answer to two decimal places.
(c) Let F(t) be the number of foxes in the region t years after their introduction. If
dF
= 0.7F(t),
dt
find the time at which the rate of decrease of the rabbit population is equal to
the rate of increase of the fox population, correct to two decimal places.
dR
dF
Hint. Note that
and
represent the rates of change of the rabbit and fox
dt
dt
populations respectively.
(d) Identify any problems with this model.

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