The population on an island is represented by the function  p(t) = 5000/1+ 12e^(-2t/3) t=time of year, determine the upper limit on # of deer  t=time of year, determine the upper limit on the...

The population on an island is represented by the function

p(t) = 5000/1+ 12e^(-2t/3)

t=time of year, determine the upper limit on # of deer

t=time of year, determine the upper limit on the number of deer that the island could support. explain your reasoning

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should remember that using the first derivative of the function, you may find the extreme values of function, hence, you need to use derivatives to find the upper limit such that:

`p'(t) = (-5000(1 + 12e^(-2t/3))')/((1 + 12e^(-2t)/3)^2)`

`p'(t) = -5000(-8e^(-2t/3))/((1 + 12e^(-2t)/3)^2)`

`p'(t) = (40000e^(-2t/3))/((1 + 12e^(-2t)/3)^2)`

You need to solve the equation `(40000e^(-2t/3))/((1 + 12e^(-2t)/3)^2) = 0`  and you may notice that there is no real real value of t for `p'(t) = 0` .

You also need to notice that 40000 needs to be the multiple of denominator `e^(2t/3) + 12`  such that:

`+-1 ; +-2 ; +-4;+-8; +-10; +-16;+-...;+-40000`

Hence, solving for t the equation `e^(2t/3) + 12`  needs to yield an integer number ( represents the number of deer).