$2.($a$)$ Choose$3$timest$0,$ t$1,$ andt$2,$ witht$1-$ t$0=$t$2-$ t$1$ Showthat$(3)$determinesa and b uniquely in terms of t$0,$p$($t$0),$ t$1,$p$($t$1),$ t$2,$ andp$($t$2).$

$($b$)$ Show that the period of accelerated growth for the United States ended in April, $1913.$

$($c$)$ Let a populationp$($t$)$ grow according to the logistic law $(3),$ and let t be the time at which half the limiting population is achieved. Show that

a$/$b p$($t$)=\_\_\_\_\_\_$:$\_\_\_$

$\_\_$ I$+$ e$-$a$($r$-1)$