I am working on the following problem:
In recitation a population model was studied in which the natural growth rate of the population of oryx was a constant $k > 0$, so that for small time intervals $\Delta t$ the population change $x(t + \Delta t) − x(t)$ is well approximated by $kx(t)\Delta t$. (You also studied the effect of hunting them, but in this problem we will leave that aside.) Measure time in years and the population in kilo-oryx (ko).
A mysterious virus infects the oryxes of the Tana River area in Kenya, which causes the growth rate to decrease as time goes on according to the formula $k(t) = \dfrac{k_0}{(a + t)^2}$ for $t \ge 0$, where $a$ and $k_0$ are certain positive constants.
(a) What are the units of the constant $a$ in “$a + t$,” and of the constant $k_0$?
The problem set gives the following solution to a):
The growth rate $k(t)$ has units years$^{−1}$ (so that $k(t)x(t)\Delta t$ has the same units as $x(t)$). The variable $t$ has units years, so the $a$ added to it must have the same units, and $k_0$ must have units years in order for the units of the fraction to work out.
The part about $a$ and $t$ having the same units makes sense. However, I have doubts about the correctness of this solution. The solution itself says that $k(t)$ has units years$^{−1}$. If $k_0 = k(0)$ ($k(t)$ where $t = 0$), then is it not inconsistent to then say it has units years in the equation $k(t) = \dfrac{k_0}{(a + t)^2}$? Should it not still have units years$^{−1}$?
I would greatly appreciate it if people could please take the time to clarify this.
$k_0\neq k(t)|_{t=0}.$
First, one can check this by plugging in 0 into the expression:
$k(0)=\frac{k_0}{a^2}\neq k_0.$
So we conclude that $k_0$ does not represent $k$ at $t=0.$ $k_0$ can be understood as a constant that does not have any "meaning" by itself but rather as a constant used to create a formula, attached to whatever units make the formula work.
I am not knowledgeable at all about this, but perhaps a similar example would be the Gravitational Constant.