For The Following Exercises, Use A Graphing Calculator And The Given Scenario: The Population Of A Fish Farm Over Time Is Modeled By The Equation $P(t) = \frac{1000}{1 + 9e^{-0.0 T}}$.17. Graph The Function.18. What Is The Initial Population

by ADMIN 242 views

Understanding the Population Growth Equation

The population of a fish farm over time is modeled by the equation P(t)=10001+9e−0.17tP(t) = \frac{1000}{1 + 9e^{-0.17t}}. This equation represents the population of the fish farm at any given time tt, where tt is measured in units of time (e.g., days, weeks, months). The equation is a type of logistic growth model, which is commonly used to describe population growth in biological systems.

Graphing the Function

To graph the function, we can use a graphing calculator. First, we need to enter the equation into the calculator. We can do this by typing in the equation and using the calculator's built-in functions to evaluate the expression.

P(t) = 1000 / (1 + 9 * exp(-0.17 * t))

Once we have entered the equation, we can graph the function by plotting the values of P(t)P(t) against the values of tt. We can use the calculator's built-in graphing features to customize the graph and add labels and titles.

Initial Population

To find the initial population, we need to evaluate the function at t=0t = 0. This is because the initial population is the population at the beginning of the time period, which is when t=0t = 0.

P(0) = 1000 / (1 + 9 * exp(-0.17 * 0))
P(0) = 1000 / (1 + 9 * 1)
P(0) = 1000 / 10
P(0) = 100

Therefore, the initial population is 100.

Interpreting the Graph

The graph of the function shows the population of the fish farm over time. The graph is a sigmoid curve, which is characteristic of logistic growth models. The curve starts at the initial population of 100 and grows rapidly at first, but then slows down as the population approaches its carrying capacity.

Carrying Capacity

The carrying capacity is the maximum population that the fish farm can support. In this case, the carrying capacity is 1000, which is the value of the function when tt approaches infinity.

lim t→∞ P(t) = 1000 / (1 + 9 * exp(-0.17 * ∞))
lim t→∞ P(t) = 1000 / (1 + 9 * 0)
lim t→∞ P(t) = 1000 / 1
lim t→∞ P(t) = 1000

Therefore, the carrying capacity is 1000.

Half-Life

The half-life is the time it takes for the population to reach half of its carrying capacity. In this case, the half-life is the time it takes for the population to reach 500.

P(t) = 1000 / (1 + 9 * exp(-0.17 * t))
500 = 1000 / (1 + 9 * exp(-0.17 * t))
(1 + 9 * exp(-0.17 * t)) = 2
9 * exp(-0.17 * t) = 1
exp(-0.17 * t) = 1/9
-0.17 * t = ln(1/9)
t = -ln(1/9) / 0.17
t ≈ 4.03

Therefore, the half-life is approximately 4.03 units of time.

Conclusion

Q: What is the purpose of the population growth equation?

A: The population growth equation is used to model the population of a fish farm over time. It helps us understand how the population changes as time passes and how it is affected by various factors such as growth rate, carrying capacity, and initial population.

Q: What is the initial population, and how is it calculated?

A: The initial population is the population at the beginning of the time period, which is when t=0t = 0. It is calculated by evaluating the function at t=0t = 0. In this case, the initial population is 100.

Q: What is the carrying capacity, and how is it calculated?

A: The carrying capacity is the maximum population that the fish farm can support. It is calculated by evaluating the function as tt approaches infinity. In this case, the carrying capacity is 1000.

Q: What is the half-life, and how is it calculated?

A: The half-life is the time it takes for the population to reach half of its carrying capacity. It is calculated by finding the time it takes for the population to reach 500. In this case, the half-life is approximately 4.03 units of time.

Q: How does the population growth equation relate to real-world applications?

A: The population growth equation has many real-world applications, such as modeling population growth in biological systems, predicting population trends, and understanding the impact of environmental factors on population growth.

Q: Can the population growth equation be used to model population decline?

A: Yes, the population growth equation can be used to model population decline. By adjusting the growth rate and carrying capacity, we can model a population that is declining over time.

Q: How can the population growth equation be used to inform conservation efforts?

A: The population growth equation can be used to inform conservation efforts by providing insights into the population dynamics of a species. By understanding how the population is changing over time, conservationists can develop effective strategies to protect and manage the population.

Q: Can the population growth equation be used to model population growth in other contexts?

A: Yes, the population growth equation can be used to model population growth in other contexts, such as modeling population growth in cities, predicting population trends in different regions, and understanding the impact of economic factors on population growth.

Q: What are some limitations of the population growth equation?

A: Some limitations of the population growth equation include:

  • It assumes a constant growth rate, which may not be realistic in all situations.
  • It assumes a fixed carrying capacity, which may not be realistic in all situations.
  • It does not take into account other factors that may affect population growth, such as environmental factors, disease, and predation.

Q: How can the population growth equation be improved?

A: The population growth equation can be improved by:

  • Incorporating more realistic assumptions about growth rate and carrying capacity.
  • Adding more variables to the equation to account for other factors that may affect population growth.
  • Using more advanced mathematical techniques to model population growth.

Conclusion

In conclusion, the population growth equation is a powerful tool for modeling population growth in a fish farm. It provides insights into the population dynamics of the species and can be used to inform conservation efforts. However, it has some limitations, and it can be improved by incorporating more realistic assumptions and adding more variables to the equation.