A certain species of fish in a lake reproduces at a rate that doubles its population every $24~\text{hours}$. The initial population of the fish was $200$. If the population $t$ hours after the initial observation is represented by the function $p\left(t\right)=200\cdot{2^{\frac{t}{24}}}$, how long will it take for the fish population to reach $4000$?