Applied integration question?

Question

A population of 50 deer are introduced into a state park. Biologist know that the rate of growth over time of the population of the deer is given by the function: #r(t) =500/(2+t^2)# deer/year

Find the function p(t) which gives the total population of the deer after t years?

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Answer 1 · 2017-05-16T16:39:29

#p(t) = 250sqrt2tan^-1(sqrt2/2t) + 50#

Translating the words into equations we are given the following:

#(d(p(t)))/dt = 500/(2+t^2); p(0) = 50#

Use the separation of variables method:

#d(p(t)) = 500/(2+t^2)dt; p(0) = 50#

Integrate both sides:

#intd(p(t)) = int500/(2+t^2)dt; p(0) = 50#

#p(t) = 250sqrt2tan^-1(sqrt2/2t) + C; p(0) = 50#

Evaluate at #t = 0#, to find the value of C:

#50 =250sqrt2tan^-1(sqrt2/2(0)) + C#

#C = 50#