Problem 61
Question
Bacterial Growth Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N(0) 2^{t} $$ where \(N(0)\) is the population size at time \(0 .\) Find the rate of growth \(d N / d t .\) Express your solution in terms of \(N(t) .\) Show that the growth rate of the population is proportional to the population size.
Step-by-Step Solution
Verified Answer
The growth rate is \( \frac{dN}{dt} = N(t) \ln(2) \), showing it is proportional to the population size.
1Step 1: Understanding the Equation
The given equation for the bacterial population is \( N(t) = N(0) \cdot 2^t \). This represents the population size at any time \( t \), with \( N(0) \) being the initial population size. The challenge is to find the growth rate which is the derivative \( \frac{dN}{dt} \) and express it in terms of \( N(t) \).
2Step 2: Differentiate the Population Formula
To find \( \frac{dN}{dt} \), differentiate \( N(t) = N(0) \cdot 2^t \) with respect to \( t \). Use the chain rule and the fact that the derivative of \( a^t \) with respect to \( t \) is \( a^t \ln(a) \). So, \( \frac{d}{dt} (2^t) = 2^t \ln(2) \).
3Step 3: Applying the Derivative
Differentiate: \[ \frac{dN}{dt} = N(0) \cdot \frac{d}{dt} (2^t) = N(0) \cdot 2^t \ln(2) \]. Since \( N(t) = N(0) \cdot 2^t \), we substitute \( N(t) \) into the derivative: \[ \frac{dN}{dt} = N(t) \ln(2) \].
4Step 4: Show Proportionality
The expression \( \frac{dN}{dt} = N(t) \ln(2) \) shows that \( \frac{dN}{dt} \) is proportional to \( N(t) \) because \( \ln(2) \) is a constant. Therefore, the growth rate \( \frac{dN}{dt} \) is directly proportional to the population size \( N(t) \).
Key Concepts
Bacterial GrowthDerivativesPopulation DynamicsExponential Growth
Bacterial Growth
Bacterial growth refers to the increase in the number of bacterial cells, not the size of individual cells. Understanding this growth is crucial in fields like microbiology and medicine. Bacteria grow rapidly in favorable conditions, often described using exponential growth models.
During exponential growth, each bacterial cell divides to produce two cells, which then divide to produce four cells, and so on. This pattern leads to a quick increase in total population. The equation given in the problem, \( N(t) = N(0) \, 2^t \), is a typical representation of this rapid growth pattern.
During exponential growth, each bacterial cell divides to produce two cells, which then divide to produce four cells, and so on. This pattern leads to a quick increase in total population. The equation given in the problem, \( N(t) = N(0) \, 2^t \), is a typical representation of this rapid growth pattern.
- \( N(0) \) represents the initial number of bacteria.
- \( 2^t \) reflects the doubling nature of bacterial divisions.
Derivatives
In calculus, derivatives are fundamental tools for understanding how functions change. They measure the rate at which a quantity changes over time, making them essential in analyzing biological processes like bacterial growth.
For a function given as \(N(t) = N(0) \, 2^t\), taking the derivative with respect to time \(t\) helps us find how fast the bacterial population is growing at any time.
Derivatives thus provide a quantitative understanding of growth dynamics, enabling us to predict future changes.
For a function given as \(N(t) = N(0) \, 2^t\), taking the derivative with respect to time \(t\) helps us find how fast the bacterial population is growing at any time.
- The derivative \( \frac{dN}{dt} \) represents the growth rate of the population.
- Using the chain rule, if \( f(t) = a^t \), then \( \frac{d}{dt}(a^t) = a^t \ln(a)\).
Derivatives thus provide a quantitative understanding of growth dynamics, enabling us to predict future changes.
Population Dynamics
Population dynamics refers to the study of how and why populations change in size and structure over time. In the context of bacterial growth, it involves understanding the processes that allow bacteria to multiply or decline.
Analyzing population dynamics helps us understand:
In the given bacterial model, the population grows exponentially in ideal conditions, as demonstrated by the equation \( N(t) = N(0) \, 2^t \). Such dynamics are characterized by an initial rapid growth phase followed by slower growth or stabilization as resources become limited.
Population dynamics are key in ecology, fisheries, pest control, and any field requiring population management.
Analyzing population dynamics helps us understand:
- Influences of environmental factors on growth rates.
- The impact of resource availability and competition.
In the given bacterial model, the population grows exponentially in ideal conditions, as demonstrated by the equation \( N(t) = N(0) \, 2^t \). Such dynamics are characterized by an initial rapid growth phase followed by slower growth or stabilization as resources become limited.
Population dynamics are key in ecology, fisheries, pest control, and any field requiring population management.
Exponential Growth
Exponential growth describes a situation where the rate of growth is proportional to the current value, leading to faster and faster increases over time. It's a common model for bacterial growth because each bacterium gives rise to a pair of offspring in every division cycle.
Mathematically, exponential growth can be represented by equations like \( N(t) = N(0) \, e^{kt} \), where:
In our specific bacterial growth model, \( 2^t \) represents a doubling pattern, which is a type of exponential growth. This pattern is unsustainable long-term due to eventual resource limitations but is a good approximation for short-term growth.
Understanding exponential growth is essential for accurately modeling and predicting the behavior of populations in various scientific fields.
Mathematically, exponential growth can be represented by equations like \( N(t) = N(0) \, e^{kt} \), where:
- \( N(t) \) is the population at time \( t \).
- \( N(0) \) is the initial population.
- \( k \) is the growth rate constant.
In our specific bacterial growth model, \( 2^t \) represents a doubling pattern, which is a type of exponential growth. This pattern is unsustainable long-term due to eventual resource limitations but is a good approximation for short-term growth.
Understanding exponential growth is essential for accurately modeling and predicting the behavior of populations in various scientific fields.
Other exercises in this chapter
Problem 61
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