The rabbit population of an island is given by

P$($t$)=5000/(1+100$e$^-$t$)$

where e is the mumber of years since your $1800.$ This function increases slowly at first, then quickly. then slowly again towards a limiting population of $5000($you are welcome to use software to graph this function to get a feeling for what it looks like$).$

$($a$)(2$ marks$)$ Provide a real$-$world explanation for why this type of population growth $($slow$,$ then fast, then slow up to a limiting value$)$ might be realistic. Write using

$($b$)16$ marks! In which yen was the rabbit population growing fastest? Justify your answer algebraically $($e$.$g$.$ show that the population is growing fostest, not slowest, at the time you compute$).$

$($Please read. this question carefully: it is not asking yon to maximize P$().$ The algebra in the infermediate steps here can get quite ugly; persevere $($and factor$/$simplify$)!)$