Bacterial Growth Rate Calculation

This solution calculates the per-capita growth rate ($r$) for a bacterial population experiencing exponential growth over a specific time period.

Exponential Growth Model

Bacterial population growth, when not limited by resources, can be modeled using the exponential growth formula:

$ N(t) = N_0 e^{rt} $

Where:

• $N(t)$ represents the population size at time $t$.
• $N_0$ is the initial population size.
• $r$ is the per-capita growth rate (per unit time).
• $t$ is the time elapsed.
• $e$ is the base of the natural logarithm.

Calculating Per-Capita Rate

The problem provides the following information:

• Initial population, $N_0 = 10^6$ cells.
• Final population, $N(t) = 5.5 \times 10^7$ cells.
• Time interval, $t = 20$ minutes.

To find the per-capita growth rate ($r$), we substitute these values into the exponential growth equation:

$ 5.5 \times 10^7 = 10^6 e^{r \times 20} $

First, isolate the exponential term by dividing both sides by the initial population ($N_0$):

$ \frac{5.5 \times 10^7}{10^6} = e^{20r} $

Simplify the left side:

$ 55 = e^{20r} $

Next, take the natural logarithm ($\ln$) of both sides to solve for the exponent:

$ \ln(55) = \ln(e^{20r}) $

Using the property $\ln(e^x) = x$, we get:

$ \ln(55) = 20r $

Now, solve for $r$ by dividing by 20:

$ r = \frac{\ln(55)}{20} $

Using a calculator, $\ln(55) \approx 4.00733$. Substitute this value:

$ r \approx \frac{4.00733}{20} $

$ r \approx 0.2003665 \text{ per minute} $

Rounding the result to two decimal places, as required:

$ r \approx 0.20 \text{ per minute} $

Therefore, the per-capita growth rate of the bacteria is approximately 0.20 per minute.