Problem 16 - Calculus for Biology and Medicine

Question

A strain of bacteria reproduces asexually every 42 minutes. That is, every 42 minutes, each bacterial cell splits into two cells. If, initially, there is 1 bacterium, how long will it take until there are 512 bacteria?

Step-by-Step Solution

Answer

It takes 378 minutes for the bacteria to grow to 512 cells.

Step 1: Understand the Doubling Effect

The bacteria double in number every 42 minutes. This means that if you start with a certain number of bacteria, in 42 minutes, that number will double.

Step 2: Express Growth in a Mathematical Equation

If we start with 1 bacterium, after 1 doubling period (42 minutes), we have 2 bacteria. We can express this mathematically as: Number of bacteria = \(2^n\), where \(n\) is the number of doubling periods that have occurred.

Step 3: Set Up the Equation for Desired Bacteria Count

In this problem, we want to know when the bacteria count will reach 512. Using our formula, we set up the equation: \(2^n = 512\).

Step 4: Solve for the Number of Doubling Periods

To find \(n\), we solve the equation \(2^n = 512\). Since \(512 = 2^9\), we find that \(n = 9\).

Step 5: Convert Doubling Periods to Total Time

Each doubling period represents 42 minutes. Therefore, the total time is \(9 \times 42\) minutes.

Step 6: Calculate the Total Time

The calculation is \(9 \times 42 = 378\) minutes. Therefore, it takes 378 minutes to reach 512 bacteria.

Key Concepts

Doubling Time

Doubling time is a critical concept in understanding exponential growth. It refers to the period it takes for a quantity to double in size or amount. In the context of our exercise, we are dealing with bacteria, which double their count every 42 minutes. Why is this important? Because knowing the doubling time helps us predict how quickly a population grows.

Consider this: if you start with one bacterium and know it doubles every 42 minutes, you can calculate how many bacteria you'll have after several doublings. It's like a snowball effect - it starts small, but growth rapidly accelerates.

• If you know the initial quantity, you can use this formula: Quantity after n doublings = Initial quantity × 2^n.
• Doubling time is constant for processes that follow this type of growth, like bacteria or investing money with compound interest.

Understanding doubling time equips you with the tools to project future growth rates. It simplifies complex growth patterns into manageable calculations.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. While exponentials involve repeated multiplication, logarithms help us discover how many times multiplication is performed to achieve a specific value. In our exercise, we use a property of logarithms: the equation\[2^n = 512\]To solve for the number of doublings ( n), we can express this using a logarithm: \[ n = \log_2{512}\]This tells us the power we need to raise 2 to get 512.

Key benefits of understanding logarithmic functions include:

• They transform multiplication into addition, making complex calculations easier.
• They are essential in fields such as computer science, wherever exponential growth occurs.
• In financial calculations, logarithms help determine the time to achieve a growth target with constant growth rates.

When faced with any situation where growth is expressed exponentially, logarithms serve as a powerful tool to dissect and understand that growth.

Bacterial Growth

Bacterial growth is a classic example of exponential growth. Each bacterium splits into two, leading to a rapid increase in population over time. This growth model shows how life can spread under optimal conditions, characterized by doubling time and exponential growth curves.

In our exercise, we started with one bacterium. After every 42 minutes (doubling period), the population doubled, demonstrating the classic exponential growth pattern.
Bare essentials of bacterial growth:

• Exponential growth continues until resources become limited.
• Environmental factors and resource availability ultimately slow down this rampant growth.
• Understanding bacterial growth provides insights for fields like epidemiology, as it reflects how diseases can spread.

When discussing bacterial populations, remember the potential for rapid growth. It's a vivid example of how exponential growth can apply in biological contexts, influencing everything from biomedicine to ecological balance.