Finding the Population at Specific Time Intervals

Oct 13, 2025 by windailys.com 95 views

The Population Growth of a Resort Community: A Mathematical Analysis

The population growth of a resort community is a complex phenomenon that can be modeled using mathematical equations. In this article, we will analyze the population growth of a resort community using the given function $${ P(t) = \frac{600t}{7t^2 + 9}, \quad t \geq 0 }$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 is given by $${ P(t) = \frac{600t}{7t^2 + 9}, \quad t \geq 0 }$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 all non-negative values of $t$ , which means that the population is always increasing over time.

Finding the Population at Specific Time Intervals

To find the population at specific time intervals, we need to substitute the given values of $t$ 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)}{7(0)^2 + 9} \}

Simplifying the expression, we get:

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

{ P(0) = 0 \}

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

Population at $t = 1$

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

{ P(1) = \frac{600(1)}{7(1)^2 + 9} \}

Simplifying the expression, we get:

{ P(1) = \frac{600}{16} \}

{ P(1) = 37.5 \}

Therefore, the population at $t = 1$ is $37.5$ thousand.

Population at $t = 3$

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

{ P(3) = \frac{600(3)}{7(3)^2 + 9} \}

Simplifying the expression, we get:

{ P(3) = \frac{1800}{90} \}

{ P(3) = 20 \}

Therefore, the population at $t = 3$ is $20$ thousand.

Population at $t = 8$

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

{ P(8) = \frac{600(8)}{7(8)^2 + 9} \}

Simplifying the expression, we get:

{ P(8) = \frac{4800}{511} \}

{ P(8) = 9.4 \}

Therefore, the population at $t = 8$ is $9.4$ thousand.

In this article, we analyzed the population growth of a resort community using the given function $${ P(t) = \frac{600t}{7t^2 + 9}, \quad t \geq 0 }$where ${$ t $}$ is time in months. We found the population at specific time intervals and discussed the implications of the results. The population function is defined for all non-negative values of $t$ , which means that the population is always increasing over time. The population at $t = 0$ is $0$ , while the population at $t = 1, 3,$ and $8$ is $37.5, 20,$ and $9.4$ thousand, respectively.

The results of this analysis have several implications for the resort community. Firstly, the population is always increasing over time, which means that the community is growing. Secondly, the population at $t = 0$ is $0$ , which means that the community was initially empty. Finally, the population at $t = 1, 3,$ and $8$ is $37.5, 20,$ and $9.4$ thousand, respectively, which means that the community is growing at a rate of $37.5, 20,$ and $9.4$ thousand per month, respectively.

There are several future research directions that can be explored based on this analysis. Firstly, the population function can be modified to include additional factors such as birth rates, death rates, and migration rates. Secondly, the population function can be used to model the growth of other types of communities such as cities, towns, and villages. Finally, the population function can be used to analyze the impact of various policies and interventions on the growth of the community.

[1] Population Growth Models. In Encyclopedia of Mathematics, edited by Michiel Hazewinkel. Springer Netherlands, 2001.
[2] Mathematical Modeling of Population Growth. In Mathematical Modeling, edited by John H. Holmes. CRC Press, 2009.
[3] Population Dynamics. In Encyclopedia of Ecology, edited by Simon A. Levin. Elsevier, 2008.
Q&A: The Population Growth of a Resort Community

In our previous article, we analyzed the population growth of a resort community using the given function $${ P(t) = \frac{600t}{7t^2 + 9}, \quad t \geq 0 }$where ${$ t $}$ is time in months. We found the population at specific time intervals and discussed the implications of the results. In this article, we will answer some of the most 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 given by the derivative of the population function with respect to time. To find the population growth rate, we need to differentiate the population function with respect to time.

Population Growth Rate

To find the population growth rate, we need to differentiate the population function with respect to time. Let's differentiate the population function with respect to time:

{ \frac{dP}{dt} = \frac{d}{dt} \left( \frac{600t}{7t^2 + 9} \right) \}

Using the quotient rule, we get:

{ \frac{dP}{dt} = \frac{(7t^2 + 9)(600) - (600t)(14t)}{(7t^2 + 9)^2} \}

Simplifying the expression, we get:

{ \frac{dP}{dt} = \frac{4200t + 5400 - 8400t^2}{(7t^2 + 9)^2} \}

Therefore, the population growth rate of the resort community is given by the expression above.

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

A: The maximum population of the resort community can be found by finding the maximum value of the population function. To find the maximum value of the population function, we need to find the critical points of the function.

Maximum Population

To find the maximum value of the population function, we need to find the critical points of the function. Let's find the critical points of the function:

{ \frac{dP}{dt} = \frac{4200t + 5400 - 8400t^2}{(7t^2 + 9)^2} \}

Setting the numerator equal to zero, we get:

{ 4200t + 5400 - 8400t^2 = 0 \}

Solving for $t$ , we get:

{ t = \frac{-5400 \pm \sqrt{5400^2 + 33600000}}{-16800} \}

Simplifying the expression, we get:

{ t = \frac{-5400 \pm 5400}{-16800} \}

Therefore, the critical points of the function are $t = 0$ and $t = 1$ .

To find the maximum value of the population function, we need to evaluate the function at the critical points. Let's evaluate the function at the critical points:

{ P(0) = \frac{600(0)}{7(0)^2 + 9} \}

{ P(0) = 0 \}

{ P(1) = \frac{600(1)}{7(1)^2 + 9} \}

{ P(1) = \frac{600}{16} \}

{ P(1) = 37.5 \}

Therefore, the maximum population of the resort community is $37.5$ thousand.

Q: What is the population of the resort community at $t = 10$ ?

A: To find the population of the resort community at $t = 10$ , we need to substitute $t = 10$ into the population function.

Population at $t = 10$

To find the population of the resort community at $t = 10$ , we need to substitute $t = 10$ into the population function:

{ P(10) = \frac{600(10)}{7(10)^2 + 9} \}

Simplifying the expression, we get:

{ P(10) = \frac{6000}{1009} \}

{ P(10) = 5.94 \}

Therefore, the population of the resort community at $t = 10$ is $5.94$ thousand.

Q: What is the population growth rate of the resort community at $t = 10$ ?

A: To find the population growth rate of the resort community at $t = 10$ , we need to substitute $t = 10$ into the population growth rate expression.

Population Growth Rate at $t = 10$

To find the population growth rate of the resort community at $t = 10$ , we need to substitute $t = 10$ into the population growth rate expression:

{ \frac{dP}{dt} = \frac{4200t + 5400 - 8400t^2}{(7t^2 + 9)^2} \}

Simplifying the expression, we get:

{ \frac{dP}{dt} = \frac{42000 + 5400 - 840000}{(1009)^2} \}

{ \frac{dP}{dt} = \frac{-819480}{1009^2} \}

{ \frac{dP}{dt} = -0.81 \}

Therefore, the population growth rate of the resort community at $t = 10$ is $-0.81$ thousand per month.

In this article, we answered some of the most frequently asked questions about the population growth of a resort community. We found the population growth rate, maximum population, and population at specific time intervals. We also discussed the implications of the results and provided future research directions.