Problem 12 - Calculus for Biology and Medicine

Question

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). . A strain of bacteria reproduces asexually every 30 minutes. That is, every 30 minutes, each bacterial cell splits into two cells. If, initially, there is one bacterium, find the number of bacterial cells after 1 hour, 2 hours, 3 hours, 4 hours, and 5 hours.

Step-by-Step Solution

Answer

The population formula is \( N_t = 2^{2t} \). After 1, 2, 3, 4, and 5 hours, the populations are 4, 16, 64, 256, and 1024, respectively.

Step 1: Understanding the Growth Pattern

The bacterium reproduces asexually every 30 minutes, doubling itself. Thus, every 30 minutes, the number of bacteria will be twice the previous count.

Step 2: Determine the Doubling Formula

If the initial population is 1 bacterium, then after every 30 minutes which is equivalent to 0.5 hours, the population doubles. Therefore, if the time is in hours and there are 2 doubling periods per hour, the formula is given by: \[ N_t = 1 \times 2^{2t} \] where \( t \) is the time in hours.

Step 3: Calculating Population at 1 Hour

Use the formula \( N_t = 1 \times 2^{2 \times 1} \). After 1 hour, the population is: \[ N_1 = 2^2 = 4 \]

Step 4: Calculating Population at 2 Hours

Compute using the formula \( N_t = 1 \times 2^{2 \times 2} \): \[ N_2 = 2^4 = 16 \]

Step 5: Calculating Population at 3 Hours

The population after 3 hours can be found using the formula: \[ N_3 = 1 \times 2^{2 \times 3} = 2^6 = 64 \]

Step 6: Calculating Population at 4 Hours

Substitute 4 into the formula: \[ N_4 = 1 \times 2^{2 \times 4} = 2^8 = 256 \]

Step 7: Calculating Population at 5 Hours

Finally, calculate the population after 5 hours: \[ N_5 = 1 \times 2^{2 \times 5} = 2^{10} = 1024 \]

Key Concepts

Bacteria Reproduction

Bacteria are fascinating microorganisms, primarily reproducing through a process called binary fission. This is a type of asexual reproduction. During this process, a single bacterium splits into two identical daughter cells. It is an efficient way of multiplying, which allows bacteria to increase their populations rapidly. In the given problem, the bacteria strain reproduces every 30 minutes. Each bacterium splits every half an hour. Over time, this leads to exponential growth. Here, it's important to note how quickly a single bacterium can turn into a much larger population. As a result of this rapid reproduction process, understanding the pattern helps in predicting the growth of bacterial populations in various settings. This concept is crucial in fields like microbiology, environmental science, and food safety.

Doubling Time

In exponential growth, doubling time is the period it takes for a population to double in size. For our bacterial example, the doubling time is 30 minutes. This means that every 30 minutes, the population doubles. The formula to express the population at any time is pivotal:

• The initial population, in this case, is 1.
• Every 30 minutes correspond to a doubling event.

To find the population size after a given amount of time, we use the expression: \[ N_t = N_0 \times 2^{n} \]Here, \(N_0\) is the initial number of bacteria, and \(n\) is the number of doubling intervals that have passed. Given that 1 hour contains two 30-minute intervals, we use this stepwise approach to determine the bacteria counts at various times.

Population Modeling

Population modeling allows us to predict how the number of organisms in a population can change over time. For exponential growth scenarios, it's vital to have a formula that relates the initial population and the growth pattern. In this case, the model is based on the bacteria doubling every 30 minutes.

• We start with an initial single bacterium.
• Using the formula mentioned earlier, we can calculate the population at any given hour.

For instance, to find the population after 5 hours, we determine the number of 30-minute intervals within this period, which is 10 doubling events. The formula \[ N_t = 1 \times 2^{2t} \] gives us an easy way to model the population over time. By understanding this model, we can anticipate situations where bacterial growth may become a concern. Population modeling, therefore, forms the backbone of planning in various scientific and industrial applications, helping predict noteworthy changes in living organisms' quantities.