The Population, { P $}$, In Thousands, Of A Resort Community Is Given By P ( T ) = 600 T 4 T 2 + 9 , T ≥ 0 P(t)=\frac{600t}{4t^2+9}, \, T \geq 0 P ( T ) = 4 T 2 + 9 600 T ​ , T ≥ 0 , Where { T $}$ Is Time, In Months.a) Find The Population At { T = 0, 1, 3, $}$ And [$ 8

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 $P(t)=\frac{600t}{4t^2+9}$, where $t$ is time in months. We will find the population at specific time intervals and discuss the implications of the results.

The Population Function

The population function is given by $P(t)=\frac{600t}{4t^2+9}$, where $t$ is time in months. This function represents the population of the resort community at any given time $t$. The function is defined for $t \geq 0$, which means that the population is only considered for non-negative time values.

Finding the Population at Specific Time Intervals

To find the population at specific time intervals, we need to substitute the given time values into the population function. Let's find the population at $t = 0, 1, 3,$ and $8$.

Population at $t = 0$

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

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

The population at $t = 0$ is $0$, which means that the population is initially $0$.

Population at $t = 1$

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

$$P(1) = \frac{600(1)}{4(1)^2+9} = \frac{600}{13} \approx 46.15$$

The population at $t = 1$ is approximately $46.15$ thousand people.

Population at $t = 3$

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

$$P(3) = \frac{600(3)}{4(3)^2+9} = \frac{1800}{45} = 40$$

The population at $t = 3$ is $40$ thousand people.

Population at $t = 8$

To find the population at $t = 8$, we substitute $t = 8$ into the population function:

$$P(8) = \frac{600(8)}{4(8)^2+9} = \frac{4800}{313} \approx 15.35$$

The population at $t = 8$ is approximately $15.35$ thousand people.

Discussion

The population growth of the resort community is a complex phenomenon that can be modeled using mathematical equations. The population function $P(t)=\frac{600t}{4t^2+9}$ represents the population of the resort community at any given time $t$. We found the population at specific time intervals, including $t = 0, 1, 3,$ and $8$. The results show that the population grows rapidly at first, but then slows down as time increases.

The population function can be used to make predictions about the future population of the resort community. For example, if we want to know the population at $t = 10$, we can substitute $t = 10$ into the population function:

$$P(10) = \frac{600(10)}{4(10)^2+9} = \frac{6000}{401} \approx 14.95$$

The population at $t = 10$ is approximately $14.95$ thousand people.

Conclusion

In conclusion, the population growth of a resort community can be modeled using mathematical equations. The population function $P(t)=\frac{600t}{4t^2+9}$ represents the population of the resort community at any given time $t$. We found the population at specific time intervals, including $t = 0, 1, 3,$ and $8$. The results show that the population grows rapidly at first, but then slows down as time increases. The population function can be used to make predictions about the future population of the resort community.

References

• [1] Calculus by Michael Spivak
• [2] Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Appendix

Population Function Derivation

The population function $P(t)=\frac{600t}{4t^2+9}$ can be derived using the following steps:

1. Assume that the population grows at a rate proportional to the current population.
2. Use the differential equation $\frac{dP}{dt} = kP$ to model the population growth.
3. Solve the differential equation to obtain the population function $P(t)=\frac{600t}{4t^2+9}$.

Population Function Properties

The population function $P(t)=\frac{600t}{4t^2+9}$ has the following properties:

• Domain: The domain of the population function is $t \geq 0$.
• Range: The range of the population function is $0 \leq P(t) \leq 60$.
• Continuity: The population function is continuous for all $t \geq 0$.
• Differentiability: The population function is differentiable for all $t > 0$.

Population Function Graph

The population function $P(t)=\frac{600t}{4t^2+9}$ can be graphed using the following steps:

1. Plot the population function on a coordinate plane.
2. Use a graphing calculator or software to visualize the population function.
3. Analyze the graph to understand the population growth of the resort community.
   The Population Growth of a Resort Community: Q&A =====================================================

Introduction

In our previous article, we explored the population growth of a resort community using the population function $P(t)=\frac{600t}{4t^2+9}$. We found the population at specific time intervals, including $t = 0, 1, 3,$ and $8$. In this article, we will answer some frequently asked questions about the population growth of the resort community.

Q&A

Q: What is the initial population of the resort community?

A: The initial population of the resort community is $0$, which means that the population is initially $0$.

Q: How does the population grow over time?

A: The population grows rapidly at first, but then slows down as time increases. This is because the population function $P(t)=\frac{600t}{4t^2+9}$ has a quadratic term in the denominator, which causes the population growth to slow down as time increases.

Q: What is the maximum population of the resort community?

A: The maximum population of the resort community is $60$ thousand people, which occurs when $t = 10$.

Q: Is the population function continuous and differentiable?

A: Yes, the population function $P(t)=\frac{600t}{4t^2+9}$ is continuous and differentiable for all $t \geq 0$.

Q: Can the population function be used to make predictions about the future population of the resort community?

A: Yes, the population function can be used to make predictions about the future population of the resort community. For example, if we want to know the population at $t = 15$, we can substitute $t = 15$ into the population function to obtain the predicted population.

Q: What are some limitations of the population function?

A: Some limitations of the population function include:

• The population function assumes that the population grows at a constant rate, which may not be realistic in practice.
• The population function does not take into account external factors that may affect the population, such as changes in the economy or environmental factors.
• The population function is based on a simplified model of population growth, which may not capture the complexities of real-world population dynamics.

Q: Can the population function be used to compare the population growth of different resort communities?

A: Yes, the population function can be used to compare the population growth of different resort communities. For example, if we have two resort communities with different population functions, we can compare their population growth rates and make predictions about their future populations.

Conclusion

In conclusion, the population growth of a resort community can be modeled using mathematical equations. The population function $P(t)=\frac{600t}{4t^2+9}$ represents the population of the resort community at any given time $t$. We answered some frequently asked questions about the population growth of the resort community, including the initial population, population growth over time, maximum population, continuity and differentiability, and limitations of the population function. We also discussed how the population function can be used to make predictions about the future population of the resort community and compare the population growth of different resort communities.

References

• [1] Calculus by Michael Spivak
• [2] Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Appendix

Population Function Derivation

The population function $P(t)=\frac{600t}{4t^2+9}$ can be derived using the following steps:

1. Assume that the population grows at a rate proportional to the current population.
2. Use the differential equation $\frac{dP}{dt} = kP$ to model the population growth.
3. Solve the differential equation to obtain the population function $P(t)=\frac{600t}{4t^2+9}$.

Population Function Properties

The population function $P(t)=\frac{600t}{4t^2+9}$ has the following properties:

• Domain: The domain of the population function is $t \geq 0$.
• Range: The range of the population function is $0 \leq P(t) \leq 60$.
• Continuity: The population function is continuous for all $t \geq 0$.
• Differentiability: The population function is differentiable for all $t > 0$.

Population Function Graph

The population function $P(t)=\frac{600t}{4t^2+9}$ can be graphed using the following steps:

1. Plot the population function on a coordinate plane.
2. Use a graphing calculator or software to visualize the population function.
3. Analyze the graph to understand the population growth of the resort community.