Problem 89
Question
MODELING Fruit Fly Populations A population of fruit flies is growing in such a way that each generation is 1.5 times as large as the last generation. If there were 100 flies in the first generation, write a formula for a geometric sequence that gives the population of the \(n\) th generation. Find the population of the fourth generation.
Step-by-Step Solution
Verified Answer
The population of the fourth generation is approximately 338 flies.
1Step 1: Identify the First Term
The problem states that there were 100 flies in the first generation. Thus, the first term of the geometric sequence, denoted as \( a_1 \), is 100.
2Step 2: Identify the Common Ratio
Each generation is 1.5 times as large as the last. This means the common ratio \( r \) for our geometric sequence is 1.5.
3Step 3: Write the Formula for the Geometric Sequence
The general formula for the \( n \)th term of a geometric sequence is \( a_n = a_1 \times r^{(n-1)} \). By substituting \( a_1 = 100 \) and \( r = 1.5 \), the formula becomes \( a_n = 100 \times (1.5)^{(n-1)} \).
4Step 4: Calculate the Population of the Fourth Generation
Using the formula \( a_n = 100 \times (1.5)^{(n-1)} \), substitute \( n = 4 \) to find the fourth generation: \[ a_4 = 100 \times (1.5)^{3} \].
5Step 5: Compute the Expression
Calculate \( (1.5)^3 = 1.5 \times 1.5 \times 1.5 = 3.375 \). Multiply this result by 100 to find \( a_4 \): \( a_4 = 100 \times 3.375 = 337.5 \). Since the population must be a whole number, round to the nearest integer.
Key Concepts
Exponential Growth in PopulationsUnderstanding the Common RatioEffective Problem-Solving TechniquesPopulation Modeling for Practical Applications
Exponential Growth in Populations
Exponential growth is a concept where quantities grow at a consistent relative rate. This means that the growth rate continually increases in proportion to the growing total number.
In the context of our fruit fly population, each new generation is 1.5 times the size of the previous generation. This is a classic example of exponential growth.
Each time you calculate the new population, it builds on the previous one at an increasing rate. This can lead to rapid growth if unchecked, as is often modeled in theoretical scenarios.
Exponential growth is commonly seen in natural settings such as populations of organisms in an ideal environment where resources are unlimited.
In the context of our fruit fly population, each new generation is 1.5 times the size of the previous generation. This is a classic example of exponential growth.
Each time you calculate the new population, it builds on the previous one at an increasing rate. This can lead to rapid growth if unchecked, as is often modeled in theoretical scenarios.
Exponential growth is commonly seen in natural settings such as populations of organisms in an ideal environment where resources are unlimited.
Understanding the Common Ratio
The common ratio is a crucial part of sequences, particularly geometric ones. It is the factor by which we multiply each term to get the next term in the sequence.
- In our fruit fly example, the common ratio is 1.5. This means every new generation is 1.5 times the population of the previous generation.
- The value of the common ratio influences how fast the sequence grows or shrinks.
- If the common ratio is greater than 1, the sequence represents growth, as in this exercise. If it is less than 1, the sequence would represent a decline.
Effective Problem-Solving Techniques
Solving problems effectively involves breaking down the tasks into manageable steps. This not only makes the problem easier to handle but also reduces the chance of errors.
In the exercise, the process started with identifying known values like the first term and the common ratio. Then, it progressed to forming a formula for the sequence.
By calculating each part step by step, such as determining the population of each generation through replacement and computation, the problem becomes clearer and easier to solve.
Developing such a structured approach to problem-solving is beneficial in tackling various mathematical and real-world problems.
In the exercise, the process started with identifying known values like the first term and the common ratio. Then, it progressed to forming a formula for the sequence.
By calculating each part step by step, such as determining the population of each generation through replacement and computation, the problem becomes clearer and easier to solve.
Developing such a structured approach to problem-solving is beneficial in tackling various mathematical and real-world problems.
Population Modeling for Practical Applications
Population modeling is a useful tool for understanding how populations change over time. It helps scientists and researchers predict future changes and organize resources accordingly.
- In our example, modeling the fruit fly population uses a geometric sequence to represent the increase in population size over generations.
- This approach can apply to other species or entities, such as bacteria growing in a petri dish or even investment growth over time.
- While the model is simplistic, it provides valuable insights that can guide real-world decisions or further research.
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