SCalcET$93\mathrm{.}8\mathrm{.}002.$

A common inhabitant of human intestines is the bacterium Escherichia coll, named after the German pediatrician Theodor Escherich, who identified it in $1885.$ A cell of this bacterium in a nu nt$-$broth medium divides into two cells every $20$ minutes. The initial population of a culture is $50$ cells.

$($a$)$ Find the relative growth rate.

$k=h{r}^{-1}$

$($b$)$ Find an expression for the number of cells after $t$ hours.

$P(t)=$

$$

$($c$)$ Find the number of cells after $7$ hours.

$$ cells

$($d$)$ Find the rate of growth after $7$ hours. $($Round your answer to the nearest integer.$)$

$$ cells$/$h

$($e$)$ When will the population reach a million cells? $($Round your answer to two decimal places.$)$

$$ h