Problem 16 - Algebra 2

Question

For Exercises 15 and \(16,\) use the following information. Bacteria usually reproduce by a process known as binary fission. In this type of reproduction, one bacterium divides, forming two bacteria. Under ideal conditions, some bacteria reproduce every 20 minutes. Write the equation for modeling the exponential growth of this bacterium.

Step-by-Step Solution

Answer

The equation is \(N(t) = N_0 \times 2^{t/20}\).

Step 1: Understand Binary Fission

Binary fission is a process where a single bacterial cell splits into two identical daughter cells. This means the population of bacteria doubles with each division under ideal conditions.

Step 2: Determine Growth Time

Bacteria that reproduce through binary fission under ideal conditions divide every 20 minutes. This means that in one hour, there are three cycles of division (since there are 60 minutes in an hour, and the fission occurs every 20 minutes).

Step 3: Define the Initial Conditions

Let the initial number of bacteria at time zero be denoted as \( N_0 \). This will be the starting point for our exponential growth model.

Step 4: Develop the Exponential Growth Equation

The exponential growth of the bacteria can be modeled by the equation:\[N(t) = N_0 \times 2^{t/20}\]Here, \(t\) is the time in minutes, and the exponent \(t/20\) indicates how many 20-minute intervals have passed, which corresponds to how many times the population doubles.

Step 5: Write the Final Equation

To summarize, the exponential growth equation that models the bacteria's growth over time is:\[N(t) = N_0 \times 2^{t/20}\]This equation reveals the relationship between time (in minutes) and the population size of the bacteria.

Key Concepts

Binary Fission

Binary fission is a fascinating method of bacterial reproduction. It is a simple and efficient process where one bacterial cell divides into two identical daughter cells. This type of reproduction allows for rapid population growth, particularly under ideal conditions. Bacteria equipped with the ability to undergo binary fission can quickly multiply, making this mechanism incredibly successful in nature.

• Binary fission starts when the parent bacterium replicates its DNA.
• Next, the cell elongates, and the DNA copies separate as the cell prepares to divide.
• Finally, the cell membrane pinches inwards, dividing the parent into two new cells.

These new cells are genetically identical to the original bacterium, which implies that under consistent environmental conditions, each division cycle results invariably in population doubling.

Bacterial Reproduction

Bacteria predominantly reproduce through binary fission, which is a form of asexual reproduction. This form of reproduction is characterized by its simplicity and speed as it does not require the complex processes involved in sexual reproduction. The ability of bacteria to reproduce rapidly through binary fission is an essential factor for their survival and adaptability in diverse environments. Here’s why bacterial reproduction through binary fission is significant:

• It's a quick way to increase population size, enabling bacteria to exploit available resources efficiently.
• It does not require a mate, making it highly advantageous in isolated environments.
• The simplicity of the process minimizes errors during reproduction, as there are fewer steps involved.

Due to these characteristics, bacterial populations can expand significantly over short periods, provided that environmental conditions are favorable and resources are available.

Population Doubling

Population doubling refers to the process by which a population reaches a size that is double its original amount. In the context of bacterial growth through binary fission, this occurs every time a bacterium divides. Given that many types of bacteria can reproduce every 20 minutes under optimal conditions, they undergo population doubling three times an hour. This rapid increase can be modeled as a sequence of exponential growth steps:

• Initial population starts with a single bacterium.
• After 20 minutes: Population doubles to 2.
• After 40 minutes: Population doubles again to 4.
• After 60 minutes (1 hour): Population reaches 8, doubling for the third time.

This continuous doubling is characteristic of exponential growth, where the population size grows at a rate proportional to its current size. Each doubling cycle builds upon the population from the previous cycle, demonstrating just how quickly bacteria can multiply.

Growth Model Equation

The growth model equation used to describe the exponential growth of bacteria through binary fission is a powerful tool in understanding bacterial population dynamics. The equation \[ N(t) = N_0 \times 2^{t/20} \] is designed to calculate the population size ( N(t) ) at any particular time ( t ), based on the initial population size ( N_0 ).Key elements of this equation include:

• Initial Population ( N_0 ): Represents the number of bacteria at the start of the observation period.
• Doubling Constant (20 minutes): Accounts for the time required for one doubling cycle under ideal conditions.
• Time ( t ): Measured in minutes to determine how many doubling cycles have occurred.

This equation is particularly useful in microbiology and environmental studies, allowing scientists to predict how fast a bacterial population will grow under given conditions. It emphasizes the impact of time on bacterial growth when resources are limited and environmental conditions are optimal.