The problem statement is as below.
We will handle the incubator with the microbes inside of it.
The microbe increases consuming a nutrition.
The amount of the nutrition is never increased.
$10^3*P(t):=$ The number of the microbes inside of the incubator at time $t$[h].
$f(t):=$ The amount of the nutrition inside of the incubator.
$P(t):=P_0*$exp$(k*t);$
$k:=0.1386;$
$P_0:=$The return value of $P(t)$ as time $t=0$;
$c\Delta t:=$The amount of the nutrition which is consumed at micro time $\Delta t[h]$ by the $50*10^3$ microbes.
Represent the differential equation of the change of the value of $f(t)$ as parameter $t$ changes with using $c$ ,$P(t)$ .
Here is how I look at it -
Number of microbes at time $t = 10^3 P(t)$
$f(t)$ is the amount of the nutrition at time $t$.
$c \Delta t$ is the amount of nutrition consumed in time $\Delta t$ by $50 \times 10^3$ microbes.
So amount of nutrition consumed in $\Delta t$ at time $t = \displaystyle \frac{c \Delta t}{50 \times 10^3} \times 10^3 \times P(t)$
$\Delta f(t) = \displaystyle \frac{c}{50} P(t) \Delta t$
lead to $\displaystyle \frac{df(t)}{dt} = \frac{c}{50} P(t) \, \, $ (in terms of $c, P(t)$)
If you have to find in terms of $P_0$ replace $P(t)$ with its formula in terms of $P_0$ and $t$.