Problem 5
Question
In Problems 5-7, give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=2\); population doubles every 20 minutes; one unit of time is 20 minutes
Step-by-Step Solution
Verified Answer
The formula is \(N(t) = 2^{t+1}\).
1Step 1: Understand the Problem Context
We are given an initial population size \(N_0 = 2\) and the population doubles every 20 minutes. We need to find a formula for \(N(t)\), where \(t = 0, 1, 2, \ldots\), considering one unit of time is 20 minutes.
2Step 2: Recognize the Formula Type
Since the population doubles every 20 minutes, this is a classic example of exponential growth. The population at each time interval is multiplied by 2.
3Step 3: Develop the Formula
Using the exponential growth model, we recognize that \(N(t)\) is given by \(N_0 \times 2^t\), where \(N_0\) is the initial population and \(t\) is the number of time intervals.
4Step 4: Write the Specific Formula
Substitute \(N_0 = 2\) into the formula. Hence, \(N(t) = 2 \times 2^t = 2^{t+1}\).
Key Concepts
Population DoublingExponential Growth ModelInitial Population Size
Population Doubling
Population doubling is a fascinating concept often seen in exponential growth scenarios. It describes a situation where a population grows by 100% over a certain period, resulting in the doubling of its initial size. In our exercise, the population doubles every 20 minutes. This rate of growth is constant, which means no matter the size of the population, it always takes the same amount of time to double.
To understand this visually, imagine a small colony of bacteria: if you start with 2 bacteria, after the doubling period, you will have 4, then 8, then 16, and so forth. The doubling is a clear sign of exponential growth, giving us an intuitive grasp of how quickly a population can increase under constant conditions. This characteristic is crucial in many scientific fields, from biology to economics, as it helps predict future population sizes under similar consistent conditions.
To understand this visually, imagine a small colony of bacteria: if you start with 2 bacteria, after the doubling period, you will have 4, then 8, then 16, and so forth. The doubling is a clear sign of exponential growth, giving us an intuitive grasp of how quickly a population can increase under constant conditions. This characteristic is crucial in many scientific fields, from biology to economics, as it helps predict future population sizes under similar consistent conditions.
- Doubling is constant.
- Aids in predicting future growth.
- Reveals exponential nature of increase.
Exponential Growth Model
The exponential growth model is pivotal in understanding complex population dynamics as it provides a mathematical way to describe consistent growth patterns. In essence, exponential growth occurs when the rate of increase of a population is proportional to its current size, leading to growth by constant percentage intervals over equal periods. This results in a rapid, accelerating increase.
To model this, we use the formula, \(N(t) = N_0 \times 2^t\), where \(N(t)\) is the population size after \(t\) time intervals, \(N_0\) is the initial population size, and the base, 2, indicates the doubling pattern. This specific formula arises because each interval results in a doubling of the population, making it ideal for situations where the multiplication factor remains consistent over time.
To model this, we use the formula, \(N(t) = N_0 \times 2^t\), where \(N(t)\) is the population size after \(t\) time intervals, \(N_0\) is the initial population size, and the base, 2, indicates the doubling pattern. This specific formula arises because each interval results in a doubling of the population, making it ideal for situations where the multiplication factor remains consistent over time.
- Describes rapid growth.
- Depends on current population size.
- Models consistent growth intervals.
Initial Population Size
Every exponential growth model begins with an initial population size, an essential component that defines the starting point of the population. In our scenario, the initial population size \(N_0\) is 2. This value is crucial as it sets the baseline for all future calculations and predicts the population size at any given point using the growth formula.
By substituting this initial value into the exponential growth equation, the formula becomes \(N(t) = 2^{t+1}\). This not only simplifies the calculation process but also highlights the exponential nature of the model, as each onward step multiplies this initial size by the growth factor, 2, raised to the power of time intervals. Understanding the impact of the initial population allows for a coherent analysis of growth trends over time.
By substituting this initial value into the exponential growth equation, the formula becomes \(N(t) = 2^{t+1}\). This not only simplifies the calculation process but also highlights the exponential nature of the model, as each onward step multiplies this initial size by the growth factor, 2, raised to the power of time intervals. Understanding the impact of the initial population allows for a coherent analysis of growth trends over time.
- Provides starting point for growth calculations.
- Essential for accurate modeling.
- Influences future population predictions.
Other exercises in this chapter
Problem 4
In Problems , produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=0.2(0.8)^{t} $$
View solution Problem 5
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Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{1}{\sqrt{n+1}} $$
View solution Problem 6
In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=4\); population doubles every 40 minutes; one unit
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