The rate of growth $d\frac{P}{d}t$ of a population of bacteria is proportional to the square root of $t,$ where $P$ is the population size

and $t$ is the time in days $(0t10)$ as shown below.

$\frac{dP}{dt}=k\sqrt[\phantom{2}]{t}$

The initial size of the population is $100.$ After $1$ day the population has grown to $400.$ Estimate the population after $8$ days.

$($Round your answer to the nearest whole number.$)$