Problem 13 - Calculus for Biology and Medicine

Question

In Problems give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. . A strain of bacteria reproduces asexually every hour. That is, every hour, 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 bacteria double each hour; after 1 hour: 2, 2 hours: 4, 3 hours: 8, 4 hours: 16, 5 hours: 32.

Step 1: Understand the Reproduction Process

This strain of bacteria doubles in number every hour because each bacterium splits into two. Initially, we have 1 bacterium.

Step 2: Find the Pattern

After 1 hour, the initial bacterium will become 2. After 2 hours, those 2 bacteria will each split, resulting in 4 bacteria. After 3 hours, those 4 will each split, resulting in 8 bacteria. Continue this pattern for each hour.

Step 3: Formulate a General Expression

From the pattern, it is clear that the bacteria double each hour. Therefore, the number of bacteria after \(t\) hours is given by the formula: \[ N(t) = 2^t \].

Step 4: Calculate Specific Values

Using the formula \( N(t) = 2^t \), calculate: - After 1 hour: \( N(1) = 2^1 = 2 \)- After 2 hours: \( N(2) = 2^2 = 4 \)- After 3 hours: \( N(3) = 2^3 = 8 \)- After 4 hours: \( N(4) = 2^4 = 16 \)- After 5 hours: \( N(5) = 2^5 = 32 \).

Key Concepts

Bacterial Reproduction

Bacterial reproduction, especially asexual reproduction, happens through a process known as binary fission. In this process, each single bacterium divides into two new bacteria. This method is incredibly efficient and leads to exponential population growth. Each parent bacterium replicates its genetic material and divides, creating two genetically identical daughter cells.

When studying bacterial growth, time intervals are often pivotal. For example, certain bacteria, like in our given exercise, can double their numbers every hour. Starting with just one bacterium, by the end of the first hour, you'll have two. This doubling means that after the second hour, those two bacteria become four, and so on. Bacteria don't just add numbers to their population; they multiply by significant amounts every cycle.

Mathematical Modeling

Mathematical modeling is key in understanding and predicting bacterial growth. It involves translating real-world situations, such as bacterial reproduction, into mathematical language. This allows us to calculate outcomes, like population sizes at different time intervals, using mathematical formulas.

In our exercise, we can describe bacterial growth using an exponential function due to the doubling nature of the reproduction process. The formula we derive is \(N(t) = 2^t\), where \(N(t)\) represents the number of bacteria after \(t\) hours. Mathematical models like this one are powerful because they provide a straightforward way to compute large numbers accurately, showing just how swiftly populations can grow and indicating trends in data. These models are vital, especially in fields like epidemiology and biology, as they help us understand potential spread rates and inform decision-making processes.

Sequence and Series

The concept of sequences and series is fundamental in breaking down patterns seen in mathematical modeling of exponential growth. A sequence is a list of numbers ordered by a specific rule, and in this case, it's defined by exponential growth rates.

The given series from our exercise, \(2, 4, 8, 16, 32\), represents the sequence of bacterial population sizes over time. Each term is generated by multiplying the previous term by 2, highlighting the exponential nature of growth. This sequence is formally expressed in mathematical terms as \([2^1, 2^2, 2^3, 2^4, 2^5]\).

Understanding sequences and series not only clarifies how populations evolve over time but also reinforces the notion of exponential versus linear growth. Such comprehension is crucial across different scientific disciplines for analysis and prediction of growth patterns in various biological processes and systems.