I solved the first 3 problems, I have problems with the fourth one. My guess would be as $t\to \infty$ we have that $N\to \infty$, and that isn't realistic.
The growth of algae in a polluted river is governed by the equation $N=N_0 e^{\alpha t},$ where $N$ is the number of organisms per unit volume of river water, $t$ is the time in weeks from the start of the observation, and $N_0$ and $\alpha$ are constants.
- After 4 weeks, the number of organisms $N$ is observed to be double the initial number. Find the value $\alpha$ to 4 significant figures.
- If $N_0=20,$ what is the value of $N$ after $10$ weeks of observation?
- How many weeks does it take for $N$ to triple its initial value?
- Give a reason why this model of pollution is unrealistic.
$\textbf{Solution:}$
- The inital number of organisms is the value of $N$ for $t=0$, in other words the initial number of organisms equals $N=N_0$ because $e^0=1$. After $t=4$ weeks we get that $N=2N_0.$ By putting that into the formula we get that $$2N_0=N_0e^{4\alpha}\Leftrightarrow 2=e^{4\alpha}\Leftrightarrow 4\alpha =\ln 2 \Leftrightarrow \alpha=\frac{1}{4}\ln 2= 0.1733$$
- For $N_0=20$ and $t=10$ we get that $$N= 20e^{10\cdot 0.1733} \approx 113.152$$
- Now we have that $N=3N_0$ and we calculated earlier that $\alpha = 0.1733,$ so we have that $$3N_0 = N_0 e^{0.1733 t}\Leftrightarrow 3=e^{0.1733 t} \Leftrightarrow 0.1733 t=\ln 3 \Leftrightarrow t= \frac{1}{0.1733}\ln 3 \approx 6.34$$
Your guess is right. It doesn't make sense that $N$ would keep growing without bound forever; only so many organisms can fit into a unit volume of water. According to the model, eventually $N$ would be larger than the number of atoms in the universe, and that can't be right.