$(a)$ Find $an$ expression for the number $of$ bacteria after $t$ hours. $(Round$ your numeric values $to$ four decimal places.$)$

$P(t)=1$

$(b)$ Find the number $of$ bacteria after $7$ hours. $(Round$ your answer $to$ the nearest whole number.$)$

$P(7)=,$bacteria

$(c)$ Find the rate $of$ growth $(in$ bacteria per hour$)$ after $7$ hours. $(Round$ your answer $to$ the nearest whole number.$)$

$P^{\text{'}}(7)=,$bacteria per hour

$(d)$ After how many hours will the population reach $250,000?(Round$ your answer $to$ one decimal place.$)$

$t=,hr$