A ship carrying $1000$ passengers has the misfortune $tobe$ wrecked $on$ a small island from which the passengers are never rescued. The natural

resources $of$ the island restrict the growth $of$ the population $to$ a $limiting$ value $of5780,to$ which the population gets closer and closer but which $it$

never reaches. The population $of$ the island after time $t,in$ years, $is$ approximated $by$ the logistic equation $P(t)=\frac{5780}{1+4\mathrm{.}78e^{-0\mathrm{.}4t}}.$

Interpret the meaning $of{P}^{\text{'}}(3)=27.$