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(e) At what time was the population growing the most rapidly? (Round your answer to two decimal places.) yr

(e) At what time was the population growing the most rapidly? (Round your answer to two decimal places.) yr

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The question I need help on is the very last question regarding the year the population grew the most rapidly.

The equation provided is: \[ N = \frac{148}{1 + 3.6e^{-2.61t}} \] where \( t \) is measured in years and \( N \) is measured in thousands of tons of fish. ### Questions and Answers (a) **What is the \( r \) value for the Eastern Pacific yellowfin tuna?** - \( r = 2.61 \) per year (b) **What is the carrying capacity \( K \) for the Eastern Pacific yellowfin tuna?** - \( K = 148 \) thousand tons (c) **What is the optimum yield level?** - 74 thousand tons (d) **Use your calculator to graph \( N \) against \( t \).** ### Description of Graph The graph is labeled as "Maple Generated Plot." It shows the population size \( N \) on the vertical axis (ranging from 0 to 200 thousand tons) against time \( t \) on the horizontal axis. The curve demonstrates a logistic growth model. Initially, the population grows rapidly, and then it levels off as it approaches the carrying capacity of 148 thousand tons.
Transcribed Image Text:The equation provided is: \[ N = \frac{148}{1 + 3.6e^{-2.61t}} \] where \( t \) is measured in years and \( N \) is measured in thousands of tons of fish. ### Questions and Answers (a) **What is the \( r \) value for the Eastern Pacific yellowfin tuna?** - \( r = 2.61 \) per year (b) **What is the carrying capacity \( K \) for the Eastern Pacific yellowfin tuna?** - \( K = 148 \) thousand tons (c) **What is the optimum yield level?** - 74 thousand tons (d) **Use your calculator to graph \( N \) against \( t \).** ### Description of Graph The graph is labeled as "Maple Generated Plot." It shows the population size \( N \) on the vertical axis (ranging from 0 to 200 thousand tons) against time \( t \) on the horizontal axis. The curve demonstrates a logistic growth model. Initially, the population grows rapidly, and then it levels off as it approaches the carrying capacity of 148 thousand tons.
The image displays four graphs representing logistic growth functions, which are used to model population growth. Each graph plots the variable \( N \) (population size) against time \( t \). ### Graph Descriptions: 1. **Top Left Graph:** - The graph shows a standard logistic growth curve. - The curve starts near the origin, rises steeply, and then begins to plateau. - This indicates rapid initial growth that slows as the population approaches the carrying capacity. 2. **Top Right Graph:** - The plot appears mostly flat. - It lacks the characteristic steep rise of logistic growth, suggesting little to no population change over time. 3. **Bottom Left Graph:** - Similar to the top left, it demonstrates a logistic curve. - The steepest part of the curve, signifying the highest growth rate, occurs early and gradually levels off. 4. **Bottom Right Graph:** - Another logistic curve, like the bottom left graph. - Markedly steep growth phase followed by a leveling off as it reaches the carrying capacity. ### Question: (e) "At what time was the population growing the most rapidly?" - This question asks for the time at which the population's growth rate is highest. This typically occurs at the inflection point of the logistic curve. ### Instruction: - Round your answer to two decimal places when determining the time of most rapid population growth. These graphs visualize how logistic functions model population growth, transitioning from exponential growth to a steady state as resources become limited.
Transcribed Image Text:The image displays four graphs representing logistic growth functions, which are used to model population growth. Each graph plots the variable \( N \) (population size) against time \( t \). ### Graph Descriptions: 1. **Top Left Graph:** - The graph shows a standard logistic growth curve. - The curve starts near the origin, rises steeply, and then begins to plateau. - This indicates rapid initial growth that slows as the population approaches the carrying capacity. 2. **Top Right Graph:** - The plot appears mostly flat. - It lacks the characteristic steep rise of logistic growth, suggesting little to no population change over time. 3. **Bottom Left Graph:** - Similar to the top left, it demonstrates a logistic curve. - The steepest part of the curve, signifying the highest growth rate, occurs early and gradually levels off. 4. **Bottom Right Graph:** - Another logistic curve, like the bottom left graph. - Markedly steep growth phase followed by a leveling off as it reaches the carrying capacity. ### Question: (e) "At what time was the population growing the most rapidly?" - This question asks for the time at which the population's growth rate is highest. This typically occurs at the inflection point of the logistic curve. ### Instruction: - Round your answer to two decimal places when determining the time of most rapid population growth. These graphs visualize how logistic functions model population growth, transitioning from exponential growth to a steady state as resources become limited.
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