Logarithm Functions — Question 9

Question 9

A bacteria culture grows according to the function $$N(t) = N_0 e^{kt}$$ where $N_0$ is the initial population, $k$ is a constant, and $t$ is the time in hours.

Suppose a culture starts with 500 bacteria and grows to 2000 bacteria in 6 hours.

• Find the growth rate $k$.

• Use your result to estimate the population after 10 hours.

• After how many hours will the population reach 10,000?

Question 9 - Solution

Growth rate. $2000=500e^{6k}$ gives

$$\boxed{k=\frac{\ln4}{6}\approx 0.231049\ \mathrm{hr}^{-1}}.$$

Population after ten hours.

$$N(10)=500\cdot4^{10/6}\approx 5039.68,$$

so the estimated whole-number population is $\boxed{5040}$.

Time to reach $10000$.

$$500e^{kt}=10000\quad\Longrightarrow\quad \boxed{t=\frac{6\ln20}{\ln4}\approx 12.97\ \text{hours}}.$$

Substitution into the exact model verifies both target populations.