Q:

Rate of growth colony of ants

The rate of growth of a colony of ants is proportional to the square root of the size of the population......

Show that if $P$ is the size of the population and $t$ is the time in days then $\dfrac{dP}{dt} = k\sqrt{P}$, where $k$ is a constant.

When $t=0$, $P=100$ and when $t=5$, $P=625$. Find an expression for $P$ as a function of $t$ and the size of the population when $t=10$.
I have integrated the equation, but I can't seem to find a single value for $k$.

A:

By definition, $\frac{dP}{dt}=k\sqrt{P}$. Then we have, $$\frac{dP}{\sqrt{P}}=kt\rightarrow2\sqrt{P}=kt+c\rightarrow P=\frac{(kt+c)^2}{4}.$$ Using the first initial condition, we have, $$100=\frac{c^2}{4}\rightarrow c=20.$$ We can use this $c$ to solve for $k$ then using the second initial condition, after which you can simply plug in the value of $t$ at which you want $P$.

