Question 684278
The population {{{P}}} of mice after t years is modeled by the function {{{P(t)= 1200/(1 +99e^(-0.4t))}}}


a) To find the initial population, plug in {{{0}}} for {{{t}}}

{{{P(0) = 1200 / (1 + 99e^(-0.4*0))}}}

{{{P(0) = 1200 / (1 + 99e^(0))}}}

 {{{P(0)= 1200 / (1 + 99)}}}

 {{{P(0)= 1200/100}}}

 {{{P(0)= 12}}}

b) To find when there will be {{{1000}}} mice, P(t) will equal {{{1000}}}

{{{1000 = 1200 / (1+ 99e ^(-0.4t))}}}

{{{1+ 99e ^(-0.4t) = 1200/1000 = 1.2}}}

{{{99e ^(-0.4t) = 1.2 - 1 = .2}}}

{{{e ^(-0.4t) = .2/99}}}

{{{-.4t = ln(.2/99)}}}

{{{t = ln(.2/99)/-.4}}}

{{{t = 15.5113}}} years


c) The maximum value of mice will be {{{1200}}}. To do this, simply graph it on a graphing calculator and find the maximum. It approaches {{{1200}}}.