Problem 91
Question
Suppose an insect population density per acre during year \(n\) can be modeled by the following recursively defined sequence. $$\begin{aligned} &a_{1}=500\\\ &a_{n}=0.8 a_{n-1} \end{aligned}$$ (a) Find the population for \(n=1,2,3\) (b) Graph the sequence for \(n=1,2,3, \ldots, 20 .\) Use the window \([0,21]\) by \([0,550] .\) Interpret the graph.
Step-by-Step Solution
Verified Answer
For n=1, 2, 3, the populations are 500, 400, and 320, respectively. The graph shows exponential decline.
1Step 1: Understanding the Recursive Formula
The given problem involves a recursive sequence \(a_n\) where \(a_1 = 500\) and \(a_n = 0.8 \times a_{n-1}\) for \(n \geq 2\). This formula indicates that each term in the sequence depends on the previous term multiplied by 0.8. Our task is to find the population for \(n=1, 2, 3\).
2Step 2: Calculate Population for n=1
Substitute \(n=1\) into the sequence. Since \(a_1=500\), the population for the first year is 500 insects.
3Step 3: Calculate Population for n=2
Use the recursive formula: \(a_2 = 0.8 \times a_1 = 0.8 \times 500 = 400\). Therefore, the population for \(n=2\) is 400 insects.
4Step 4: Calculate Population for n=3
Continue applying the recursive formula for \(n=3\): \(a_3 = 0.8 \times a_2 = 0.8 \times 400 = 320\). Thus, the population for \(n=3\) is 320 insects.
5Step 5: Graphing the Sequence from n=1 to n=20
Starting with \(a_1 = 500\), use the recursive formula to calculate each subsequent term up to \(n=20\). While graphing, the x-axis represents the year number, \(n\), and the y-axis represents the population. Notice that the population decreases exponentially, approaching zero as \(n\) increases. This results in a graph with an initial value of 500 that declines towards zero.
6Step 6: Graph Interpretation
As \(n\) increases, each year's population is 80% of the previous year's, resulting in an exponential decay. This suggests the insect population is decreasing gradually each year, possibly due to environmental factors limiting growth or survival.
Key Concepts
Exponential DecayPopulation ModelingGraph Interpretation
Exponential Decay
Exponential decay is a mathematical concept where a quantity decreases by a fixed percentage over regular intervals. This concept can be seen in the recursive sequence provided, where the insect population decreases by 20% each year. The decrease is not linear, which means it doesn't simply subtract a constant amount from each term. Rather, it reduces the current amount by a specific percentage. This results in a faster rate of decrease initially, which slows down as the remaining quantity gets smaller.
In our problem, each year's population is 80% of the previous year's, which represents an exponential decay. The mathematical expression is given by the recursive relation: \( a_n = 0.8 \times a_{n-1} \). Here, 0.8 represents the rate of retention, or how much of the population remains each year.
Understanding exponential decay helps explain why certain phenomena, such as the reduction in population seen here, occur more rapidly at first and then slow down as they progress. This understanding can be applied in various scientific and financial contexts, including radioactive decay and depreciation of assets.
In our problem, each year's population is 80% of the previous year's, which represents an exponential decay. The mathematical expression is given by the recursive relation: \( a_n = 0.8 \times a_{n-1} \). Here, 0.8 represents the rate of retention, or how much of the population remains each year.
Understanding exponential decay helps explain why certain phenomena, such as the reduction in population seen here, occur more rapidly at first and then slow down as they progress. This understanding can be applied in various scientific and financial contexts, including radioactive decay and depreciation of assets.
Population Modeling
Population modeling involves creating mathematical models to represent and forecast the growth or decline of populations over time. These models can be simple or complex, depending on the factors they consider. For the problem at hand, a simple model is used where only the initial population and a constant decay rate are considered.
The recursive formula provided, \( a_n = 0.8 \times a_{n-1} \), is an example of a population model based on exponential decay. Population modeling can be crucial for fields like ecology, where understanding changes in insect populations can help predict their impact on ecosystems.
By knowing the initial population and the decay rate, we can forecast future populations. This allows scientists and policymakers to plan interventions, such as conservation efforts or resource allocation, to address expected changes.
The recursive formula provided, \( a_n = 0.8 \times a_{n-1} \), is an example of a population model based on exponential decay. Population modeling can be crucial for fields like ecology, where understanding changes in insect populations can help predict their impact on ecosystems.
By knowing the initial population and the decay rate, we can forecast future populations. This allows scientists and policymakers to plan interventions, such as conservation efforts or resource allocation, to address expected changes.
Graph Interpretation
Interpreting graphs is a vital skill in understanding how mathematical models behave over time. When graphing the insect population modeled by a recursive sequence, we track the changes from year 1 to 20.
The x-axis of this graph typically represents time (years, in this case), while the y-axis represents the insect population. By plotting each year's population, we observe a downward curve characteristic of exponential decay, starting from 500 and approaching zero. This pattern highlights the diminishing population as each year passes, showcasing the effects of the decay rate over time.
Understanding the visual representation allows us to quickly assess trends and potential future behavior. It also helps illustrate complex concepts like exponential decay in an intuitive manner. Seeing the actual data points and the curve they form reinforces how populations shrink in an exponential model.
The x-axis of this graph typically represents time (years, in this case), while the y-axis represents the insect population. By plotting each year's population, we observe a downward curve characteristic of exponential decay, starting from 500 and approaching zero. This pattern highlights the diminishing population as each year passes, showcasing the effects of the decay rate over time.
Understanding the visual representation allows us to quickly assess trends and potential future behavior. It also helps illustrate complex concepts like exponential decay in an intuitive manner. Seeing the actual data points and the curve they form reinforces how populations shrink in an exponential model.
Other exercises in this chapter
Problem 90
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