Five thousand rabbits were introduced to an island that did not have any rabbits

Question

Five thousand rabbits were introduced to an island that did not have any rabbits on it. Over the next 12 months, the population of the rabbits was modeled by

P(t)=5+t4ln(t+1)

where P(t) is the population of rabbits in thousands and t is the months since the rabbits were put on the island. Over the first twelve months, [0,12] , what was the lowest number of rabbits on the island and what was the highest number of rabbits on the island? (Round answers to the nearest whole number.)

____ = Lowest Population of Rabbits

____= Highest Population of Rabbits

Explanation / Answer

P(t) = 5 + t - 4*ln (t + 1)

P'(t) = 0 + 1 - 4/(t + 1)

maximum or minimum number of rabbits will be at P'(t) = 0, or at the endpoints of given period.

1 - 4/(t + 1) = 0

1 = 4/(t + 1)

t + 1 = 4

t = 3

So,

when t = 3,

P(t) = 5 + 3 - 4*ln (3 + 1) = 2.4548 = 2454.8 = 2455 rabbilts

when t = 0,

P(t) = 5 + 0 - 4*ln (0 + 1) = 5000 rabbilts

when t = 12,

P(t) = 5 + 12 - 4*ln (12 + 1) = 6.740 = 6740 rabbilts

So,

lowest population of rabbits = 2455,

Highest population of rabbits = 6740

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