Question 1206832



let population og rabbits be {{{p}}}, time {{{t}}} (in years)


{{{p(t)=a*e^(b*t)}}}


if the colony begins with {{{5 }}}rabbits, {{{p(0)=5}}}

{{{5=a*e^(b*0)}}}

{{{5=a*1}}}

{{{a=5}}}


so far equation is

{{{p(t)=5*e^(b*t)}}}


if {{{5}}} years later there are {{{320}}} rabbits

{{{320=5*e^(b*5)}}}

{{{320/5=e^(b*5)}}}

{{{e^(b*5)=64}}}...take natural log of both sides

{{{ln(e^(b*5))=ln(64)}}}

{{{(b*5)ln(e)=ln(64)}}}

{{{(b*5)*1=4.1588830833596715}}}

{{{b=4.1588830833596715/5}}}

{{{b=0.8317766166719343}}}


equation is:

{{{p(t)=5*e^(0.8317766166719343*t)}}}


(a) estimate how long it takes for the population of rabbits to reach {{{1000}}} rabbits.

{{{1000=5*e^(0.8317766166719343*t)}}}

{{{e^(0.8317766166719343*t)=200}}}

{{{ln(e^(0.8317766166719343*t))=ln(200)}}}

{{{(0.8317766166719343*t)*ln(e)=ln(200)}}}

{{{0.8317766166719343*t=5.298317366548}}}

{{{t=5.298317366548/0.8317766166719343}}}

{{{t=6.37 }}}years