The Population, { P $}$, In Thousands, Of A Resort Community Is Given By ${ P(t) = \frac{600t}{4t^2 + 9}, , T \geq 0 }$where { T $}$ Is Time, In Months.a) Find The Population At { T = 0, 1, 3, $}$ And

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Introduction

The population growth of a resort community is a complex phenomenon that can be modeled using mathematical equations. In this article, we will explore the population growth of a resort community using the given function ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months. We will find the population at { t = 0, 1, 3, $}$ and discuss the implications of the results.

The Population Function

The population function is given by ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months. This function represents the population of the resort community at any given time tt.

Understanding the Function

The population function is a rational function, which means it is the ratio of two polynomials. The numerator of the function is 600t600t, which represents the rate at which the population is growing. The denominator of the function is 4t2+94t^2 + 9, which represents the carrying capacity of the resort community.

Carrying Capacity

The carrying capacity of a population is the maximum number of individuals that an environment can support. In this case, the carrying capacity is represented by the denominator of the function, 4t2+94t^2 + 9. As tt increases, the denominator increases, which means that the carrying capacity of the resort community also increases.

Population Growth Rate

The population growth rate is represented by the numerator of the function, 600t600t. As tt increases, the numerator increases, which means that the population growth rate also increases.

Finding the Population at Given Times

We are given the following times: t=0,1,3t = 0, 1, 3. We need to find the population at these times using the population function.

Population at t=0t = 0

To find the population at t=0t = 0, we substitute t=0t = 0 into the population function:

{ P(0) = \frac{600(0)}{4(0)^2 + 9} \}

Simplifying the expression, we get:

{ P(0) = \frac{0}{9} \}

{ P(0) = 0 \}

Therefore, the population at t=0t = 0 is 0.

Population at t=1t = 1

To find the population at t=1t = 1, we substitute t=1t = 1 into the population function:

{ P(1) = \frac{600(1)}{4(1)^2 + 9} \}

Simplifying the expression, we get:

{ P(1) = \frac{600}{13} \}

{ P(1) = 46.15 \}

Therefore, the population at t=1t = 1 is approximately 46.15 thousand.

Population at t=3t = 3

To find the population at t=3t = 3, we substitute t=3t = 3 into the population function:

{ P(3) = \frac{600(3)}{4(3)^2 + 9} \}

Simplifying the expression, we get:

{ P(3) = \frac{1800}{45 + 9} \}

{ P(3) = \frac{1800}{54} \}

{ P(3) = 33.33 \}

Therefore, the population at t=3t = 3 is approximately 33.33 thousand.

Discussion

The results show that the population of the resort community is increasing over time. The population at t=0t = 0 is 0, which means that the community was just established. The population at t=1t = 1 is approximately 46.15 thousand, which means that the community has grown significantly in just one month. The population at t=3t = 3 is approximately 33.33 thousand, which means that the community has continued to grow, but at a slower rate.

The results also show that the population growth rate is increasing over time. The population growth rate at t=0t = 0 is 0, which means that the community was just established and there was no population growth. The population growth rate at t=1t = 1 is approximately 46.15 thousand, which means that the community has grown significantly in just one month. The population growth rate at t=3t = 3 is approximately 33.33 thousand, which means that the community has continued to grow, but at a slower rate.

Conclusion

In conclusion, the population growth of a resort community can be modeled using the given function ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months. The results show that the population of the resort community is increasing over time and the population growth rate is also increasing over time. The results have implications for the management of the resort community, as they suggest that the community will continue to grow and that the population growth rate will continue to increase.

Recommendations

Based on the results, the following recommendations can be made:

  • The resort community should continue to grow and develop, as the population growth rate is increasing over time.
  • The community should invest in infrastructure and services to support the growing population.
  • The community should also invest in education and training programs to support the growing population and to ensure that the community has the skills and knowledge needed to support the growing population.

Limitations

The results of this study are limited by the fact that the population function is a simplification of the real-world situation. The population function assumes that the population growth rate is constant over time, which is not the case in reality. The population growth rate is likely to vary over time due to a variety of factors, including changes in the economy, changes in government policies, and changes in the environment.

Future Research

Future research should focus on developing a more realistic population function that takes into account the complexities of the real-world situation. This could involve incorporating more variables into the population function, such as changes in the economy, changes in government policies, and changes in the environment.

References

Appendix

The following appendix provides additional information that is not included in the main text.

Population Function Derivation

The population function is derived using the following steps:

  1. Define the population function: The population function is defined as ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months.
  2. Simplify the population function: The population function can be simplified by dividing both the numerator and the denominator by tt:

{ P(t) = \frac{600}{4t + \frac{9}{t}} \}

  1. Take the limit as tt approaches 0: The limit as tt approaches 0 can be taken by substituting t=0t = 0 into the population function:

{ \lim_{t \to 0} P(t) = \lim_{t \to 0} \frac{600}{4t + \frac{9}{t}} \}

Simplifying the expression, we get:

{ \lim_{t \to 0} P(t) = \frac{600}{9} \}

{ \lim_{t \to 0} P(t) = 66.67 \}

Therefore, the limit as tt approaches 0 is 66.67 thousand.

Population Growth Rate

The population growth rate is represented by the numerator of the population function, 600t600t. As tt increases, the numerator increases, which means that the population growth rate also increases.

Carrying Capacity

The carrying capacity of a population is the maximum number of individuals that an environment can support. In this case, the carrying capacity is represented by the denominator of the population function, 4t2+94t^2 + 9. As tt increases, the denominator increases, which means that the carrying capacity of the resort community also increases.

Population Function Graph

The population function can be graphed using the following steps:

  1. Plot the population function: The population function can be plotted by substituting different values of tt into the population function and plotting the resulting values.
  2. Add a title and labels: The graph can be labeled with a title and labels to make it easier to understand.
  3. Add a legend: The graph can be labeled with a legend to make it easier to understand.

The resulting graph shows that the population of the resort community is increasing over time and the population growth rate is also increasing over time.

Population Function Table

The population function can be represented in a table using the following steps:

  1. Create a table: A table can be created with columns for tt, P(t)P(t), and $P'(t
    Q&A: The Population Growth of a Resort Community =====================================================

Introduction

In our previous article, we explored the population growth of a resort community using the given function ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months. We found the population at { t = 0, 1, 3, $}$ and discussed the implications of the results. In this article, we will answer some frequently asked questions about the population growth of a resort community.

Q: What is the population growth rate of the resort community?

A: The population growth rate of the resort community is represented by the numerator of the population function, 600t600t. As tt increases, the numerator increases, which means that the population growth rate also increases.

Q: What is the carrying capacity of the resort community?

A: The carrying capacity of the resort community is represented by the denominator of the population function, 4t2+94t^2 + 9. As tt increases, the denominator increases, which means that the carrying capacity of the resort community also increases.

Q: How does the population function change over time?

A: The population function changes over time as the numerator and denominator change. As tt increases, the numerator increases, which means that the population growth rate also increases. The denominator also increases, which means that the carrying capacity of the resort community also increases.

Q: What is the population of the resort community at t=0t = 0?

A: The population of the resort community at t=0t = 0 is 0.

Q: What is the population of the resort community at t=1t = 1?

A: The population of the resort community at t=1t = 1 is approximately 46.15 thousand.

Q: What is the population of the resort community at t=3t = 3?

A: The population of the resort community at t=3t = 3 is approximately 33.33 thousand.

Q: How does the population growth rate change over time?

A: The population growth rate changes over time as the numerator of the population function changes. As tt increases, the numerator increases, which means that the population growth rate also increases.

Q: What are the implications of the population growth rate increasing over time?

A: The implications of the population growth rate increasing over time are that the resort community will continue to grow and develop. The community will need to invest in infrastructure and services to support the growing population.

Q: What are the limitations of the population function?

A: The limitations of the population function are that it assumes that the population growth rate is constant over time, which is not the case in reality. The population growth rate is likely to vary over time due to a variety of factors, including changes in the economy, changes in government policies, and changes in the environment.

Q: What are the recommendations for the resort community based on the population function?

A: The recommendations for the resort community based on the population function are that the community should continue to grow and develop, and that the community should invest in infrastructure and services to support the growing population.

Conclusion

In conclusion, the population growth of a resort community can be modeled using the given function ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months. The results show that the population of the resort community is increasing over time and the population growth rate is also increasing over time. The results have implications for the management of the resort community, as they suggest that the community will continue to grow and that the population growth rate will continue to increase.

Recommendations for Future Research

Based on the results of this study, the following recommendations for future research are made:

  • Develop a more realistic population function that takes into account the complexities of the real-world situation.
  • Incorporate more variables into the population function, such as changes in the economy, changes in government policies, and changes in the environment.
  • Use the population function to model the population growth of other communities and to make predictions about the future population growth of these communities.

References

Appendix

The following appendix provides additional information that is not included in the main text.

Population Function Derivation

The population function is derived using the following steps:

  1. Define the population function: The population function is defined as ${ P(t) = \frac{600t}{4t^2 + 9}, , t \geq 0 }$where { t $}$ is time, in months.
  2. Simplify the population function: The population function can be simplified by dividing both the numerator and the denominator by tt:

{ P(t) = \frac{600}{4t + \frac{9}{t}} \}

  1. Take the limit as tt approaches 0: The limit as tt approaches 0 can be taken by substituting t=0t = 0 into the population function:

{ \lim_{t \to 0} P(t) = \lim_{t \to 0} \frac{600}{4t + \frac{9}{t}} \}

Simplifying the expression, we get:

{ \lim_{t \to 0} P(t) = \frac{600}{9} \}

{ \lim_{t \to 0} P(t) = 66.67 \}

Therefore, the limit as tt approaches 0 is 66.67 thousand.

Population Growth Rate

The population growth rate is represented by the numerator of the population function, 600t600t. As tt increases, the numerator increases, which means that the population growth rate also increases.

Carrying Capacity

The carrying capacity of a population is the maximum number of individuals that an environment can support. In this case, the carrying capacity is represented by the denominator of the population function, 4t2+94t^2 + 9. As tt increases, the denominator increases, which means that the carrying capacity of the resort community also increases.

Population Function Graph

The population function can be graphed using the following steps:

  1. Plot the population function: The population function can be plotted by substituting different values of tt into the population function and plotting the resulting values.
  2. Add a title and labels: The graph can be labeled with a title and labels to make it easier to understand.
  3. Add a legend: The graph can be labeled with a legend to make it easier to understand.

The resulting graph shows that the population of the resort community is increasing over time and the population growth rate is also increasing over time.

Population Function Table

The population function can be represented in a table using the following steps:

  1. Create a table: A table can be created with columns for tt, P(t)P(t), and P′(t)P'(t).
  2. Substitute values of tt into the population function: Different values of tt can be substituted into the population function to find the corresponding values of P(t)P(t) and P′(t)P'(t).
  3. Add the values to the table: The values of P(t)P(t) and P′(t)P'(t) can be added to the table.

The resulting table shows that the population of the resort community is increasing over time and the population growth rate is also increasing over time.

Population Function Code

The population function can be represented in code using the following steps:

  1. Define the population function: The population function can be defined using a programming language such as Python or MATLAB.
  2. Substitute values of tt into the population function: Different values of tt can be substituted into the population function to find the corresponding values of P(t)P(t) and P′(t)P'(t).
  3. Add the values to the table: The values of P(t)P(t) and P′(t)P'(t) can be added to the table.

The resulting code shows that the population of the resort community is increasing over time and the population growth rate is also increasing over time.