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173. A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2%. If the initial population is 50 deer, what is the population of deer at any given time?

173. A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2%. If the initial population is 50 deer, what is the population of deer at any given time?

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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Solve 173 answer should be in form of the second picture i send
### Educational Content on Logistic Growth and Population Dynamics #### Logistic Equation Problems **171.** Solve the logistic equation \( P' = CP - P^2 \) for \( C = 10 \) and an initial condition of \( P(0) = 2 \). **172.** Solve the logistic equation \( P' = CP - P^2 \) for \( C = 40 \) and an initial condition of \( P(0) = 2 \). #### Applied Population Dynamics **173.** A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2%. If the initial population is 50 deer, what is the population of deer at any given time? **174.** A population of frogs in a pond has a growth rate of 5%. If the initial population is 1000 frogs and the carrying capacity is 6000, what is the population of frogs at any given time? **175.** Bacteria grow at a rate of 20% per hour in a petri dish. If there is initially one bacterium, what is the bacterial population at a given time? ### Explanation These problems explore logistic growth models, which describe how populations grow in an environment with limited resources. The carrying capacity represents the maximum population size that the environment can sustain. The growth rate affects how quickly the population approaches this limit. For questions 171 and 172, you will solve the given logistic equations with specific parameters, reflecting how theoretical populations might behave under similar conditions. For applications like 173, 174, and 175, you calculate the future population size using provided growth rates, initial conditions, and carrying capacities. These scenarios help illustrate real-world population dynamics such as animal populations in ecosystems and bacterial growth.
Transcribed Image Text:### Educational Content on Logistic Growth and Population Dynamics #### Logistic Equation Problems **171.** Solve the logistic equation \( P' = CP - P^2 \) for \( C = 10 \) and an initial condition of \( P(0) = 2 \). **172.** Solve the logistic equation \( P' = CP - P^2 \) for \( C = 40 \) and an initial condition of \( P(0) = 2 \). #### Applied Population Dynamics **173.** A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2%. If the initial population is 50 deer, what is the population of deer at any given time? **174.** A population of frogs in a pond has a growth rate of 5%. If the initial population is 1000 frogs and the carrying capacity is 6000, what is the population of frogs at any given time? **175.** Bacteria grow at a rate of 20% per hour in a petri dish. If there is initially one bacterium, what is the bacterial population at a given time? ### Explanation These problems explore logistic growth models, which describe how populations grow in an environment with limited resources. The carrying capacity represents the maximum population size that the environment can sustain. The growth rate affects how quickly the population approaches this limit. For questions 171 and 172, you will solve the given logistic equations with specific parameters, reflecting how theoretical populations might behave under similar conditions. For applications like 173, 174, and 175, you calculate the future population size using provided growth rates, initial conditions, and carrying capacities. These scenarios help illustrate real-world population dynamics such as animal populations in ecosystems and bacterial growth.
### Mathematical Expressions **Semi-Stable Condition:** \( P = 0 \) is described as semi-stable. --- **Highlighted Problems:** 171. \( P = \frac{10e^{10t}}{4 + e^{10t}} \) 173. \( P(t) = \frac{10000e^{0.02t}}{150 + 50e^{0.02t}} \) --- **Additional Information:** 177. 7 years 2 months --- ### Explanation of Graph: The graph is located below the mathematical expressions. Although only partially visible, it includes an axis labeled "P" with ticks and a section showing values ranging to 12,000. The graph likely represents data related to the functions or expressions in the highlighted problems above. Due to the limited view, detailed interpretation of the graph's content or specific data trends isn't possible. Further inspection is recommended for comprehensive analysis.
Transcribed Image Text:### Mathematical Expressions **Semi-Stable Condition:** \( P = 0 \) is described as semi-stable. --- **Highlighted Problems:** 171. \( P = \frac{10e^{10t}}{4 + e^{10t}} \) 173. \( P(t) = \frac{10000e^{0.02t}}{150 + 50e^{0.02t}} \) --- **Additional Information:** 177. 7 years 2 months --- ### Explanation of Graph: The graph is located below the mathematical expressions. Although only partially visible, it includes an axis labeled "P" with ticks and a section showing values ranging to 12,000. The graph likely represents data related to the functions or expressions in the highlighted problems above. Due to the limited view, detailed interpretation of the graph's content or specific data trends isn't possible. Further inspection is recommended for comprehensive analysis.
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