The Rate At Which People Enter An Amusement Park On A Given Day Is Modeled By The Function { E $}$ Defined By${ E(t) = \frac{15600}{(t^2 - 24t + 160)} }$The Rate At Which People Leave The Same Amusement Park On The Same Day Is

Mar 8, 2025 by ADMIN 228 views

The Rate at Which People Enter and Leave an Amusement Park

The rate at which people enter and leave an amusement park can be modeled using mathematical functions. In this article, we will explore the function that models the rate at which people enter the park and derive the function that models the rate at which people leave the park.

The function that models the rate at which people enter the park is given by:

E(t) = \frac{15600}{(t^2 - 24t + 160)}

This function is defined for all real numbers $t$ and represents the rate at which people enter the park at time $t$ .

To derive the function that models the rate at which people leave the park, we need to find the derivative of the function $E(t)$ with respect to $t$ . This will give us the rate at which people are leaving the park at time $t$ .

The Derivative of $E(t)$

Using the quotient rule of differentiation, we can find the derivative of $E(t)$ as follows:

\frac{dE}{dt} = \frac{(t^2 - 24t + 160) \cdot 0 - 15600 \cdot (2t - 24)}{(t^2 - 24t + 160)^2}

Simplifying the expression, we get:

\frac{dE}{dt} = \frac{-31200 + 624t}{(t^2 - 24t + 160)^2}

This is the function that models the rate at which people leave the park.

The Rate at Which People Leave the Park

The rate at which people leave the park is given by the function:

L(t) = \frac{-31200 + 624t}{(t^2 - 24t + 160)^2}

This function is defined for all real numbers $t$ and represents the rate at which people leave the park at time $t$ .

Graphing the Functions

To visualize the functions $E(t)$ and $L(t)$ , we can graph them using a graphing tool or software.

Graph of $E(t)$

The graph of $E(t)$ is a curve that opens downward and has a single maximum point at $t = 12$ . The graph is symmetric about the line $t = 12$ .

Graph of $L(t)$

The graph of $L(t)$ is a curve that opens upward and has a single minimum point at $t = 12$ . The graph is symmetric about the line $t = 12$ .

In this article, we have derived the function that models the rate at which people leave the amusement park using the function that models the rate at which people enter the park. We have also graphed the functions to visualize their behavior.

[1] Calculus, 3rd edition, by Michael Spivak
[2] Differential Equations, 2nd edition, by James R. Brannan and William E. Boyce

In future work, we can use the functions $E(t)$ and $L(t)$ to model the population of the amusement park over time. We can also use the functions to optimize the park's operations, such as scheduling and staffing.

The following is a list of the mathematical symbols used in this article:

$E(t)$ : the function that models the rate at which people enter the park
$L(t)$ : the function that models the rate at which people leave the park
$t$ : time
$t^2$ : the square of time
$t^3$ : the cube of time
$\frac{dE}{dt}$ : the derivative of $E(t)$ with respect to $t$
$\frac{dL}{dt}$ : the derivative of $L(t)$ with respect to $t$

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The Rate at Which People Enter and Leave an Amusement Park: Q&A

In our previous article, we explored the function that models the rate at which people enter an amusement park and derived the function that models the rate at which people leave the park. In this article, we will answer some frequently asked questions about the functions and their applications.

Q: What is the significance of the function E(t)?

A: The function E(t) represents the rate at which people enter the amusement park at time t. It is a mathematical model that helps us understand how the number of people entering the park changes over time.

Q: How is the function E(t) related to the function L(t)?

A: The function L(t) is the derivative of the function E(t) with respect to t. This means that L(t) represents the rate at which people are leaving the park at time t.

Q: What is the maximum value of the function E(t)?

A: The maximum value of the function E(t) occurs at t = 12, where E(12) = 15600 / (12^2 - 24*12 + 160) = 15600 / (144 - 288 + 160) = 15600 / 16 = 975.

Q: What is the minimum value of the function L(t)?

A: The minimum value of the function L(t) occurs at t = 12, where L(12) = (-31200 + 62412) / (12^2 - 2412 + 160)^2 = (-31200 + 7488) / (144 - 288 + 160)^2 = (-31200 + 7488) / 16^2 = (-31200 + 7488) / 256 = -23712 / 256 = -92.5.

Q: How can the functions E(t) and L(t) be used in real-world applications?

A: The functions E(t) and L(t) can be used to model the population of the amusement park over time. They can also be used to optimize the park's operations, such as scheduling and staffing.

Q: What are some potential limitations of the functions E(t) and L(t)?

A: Some potential limitations of the functions E(t) and L(t) include:

The functions assume that the rate at which people enter and leave the park is constant over time, which may not be the case in reality.
The functions do not take into account external factors that may affect the rate at which people enter and leave the park, such as weather or special events.
The functions are based on a simplified model of the park's operations and may not capture all the complexities of real-world scenarios.

Q: How can the functions E(t) and L(t) be modified to account for external factors?

A: The functions E(t) and L(t) can be modified to account for external factors by incorporating additional variables or parameters into the model. For example, the rate at which people enter the park may be affected by weather conditions, which can be represented by an additional variable in the model.

In this article, we have answered some frequently asked questions about the functions E(t) and L(t) and their applications. We have also discussed some potential limitations of the functions and how they can be modified to account for external factors.

[1] Calculus, 3rd edition, by Michael Spivak
[2] Differential Equations, 2nd edition, by James R. Brannan and William E. Boyce

In future work, we can use the functions E(t) and L(t) to model the population of the amusement park over time and optimize the park's operations. We can also explore other applications of the functions, such as modeling the rate at which people enter and leave other types of venues, such as shopping malls or restaurants.