Substituting the Expression for l

Oct 13, 2025 by miodeya.com 94 views

Introduction

The high school athletics department is planning to install a new rectangular addition to their current practice field. This new addition will not only provide the athletes with a larger space to train but also create a unique mathematical problem for the students and faculty to solve. In this article, we will delve into the mathematical analysis of the new practice field, exploring the relationship between its length and width.

The Original Field

Before we dive into the new addition, let's first consider the original field. We are given that the original field has an area of 1200 square meters. Since the area of a rectangle is given by the formula A = lw, where A is the area, l is the length, and w is the width, we can write the equation as:

1200 = lw

The New Addition

The new addition will have a length that is at least 10 meters more than twice the width of the new addition. Let's denote the width of the new addition as w. Then, the length of the new addition can be expressed as:

l = 2w + 10

The Combined Field

When the new addition is installed, the combined field will have a total area of 2400 square meters (1200 + 1200). The length and width of the combined field will be the sum of the length and width of the original field and the new addition, respectively. We can write the equation as:

2400 = (l + l')w'

where l' is the length of the new addition and w' is the width of the new addition.

Substituting the Expression for l

We can substitute the expression for l into the equation above:

2400 = (2w + 10 + 2w')w'

Simplifying the Equation

Simplifying the equation, we get:

2400 = (4w + 10 + 2w')w'

Expanding the Equation

Expanding the equation, we get:

2400 = 4ww' + 10w' + 2w'2

Rearranging the Equation

Rearranging the equation, we get:

2400 - 10w' = 4ww' + 2w'2

Factoring the Equation

Factoring the equation, we get:

(2w' + 4w - 10)(w' - 5) = 0

Solving for w'

Solving for w', we get:

w' = 5 or w' = -2.5 + 2w

The Width of the New Addition

Since the width of the new addition cannot be negative, we take w' = 5 as the only valid solution.

The Length of the New Addition

Substituting w' = 5 into the expression for l, we get:

l = 2(5) + 10 = 20

The Dimensions of the Combined Field

The dimensions of the combined field are:

Length: 20 + 20 = 40 meters Width: 5 + 20 = 25 meters

Conclusion

In conclusion, the high school athletics department's new practice field will have a length of at least 40 meters and a width of at least 25 meters. This new addition will not only provide the athletes with a larger space to train but also create a unique mathematical problem for the students and faculty to solve.

Mathematical Analysis

The mathematical analysis of the new practice field involved solving a system of equations to find the dimensions of the combined field. The solution to the system of equations was w' = 5, which represents the width of the new addition. The length of the new addition was found to be l = 20 meters.

Real-World Applications

The mathematical analysis of the new practice field has real-world applications in the field of engineering and architecture. The design of buildings, bridges, and other structures requires the use of mathematical equations to ensure that they are safe and functional.

Future Research

Future research in this area could involve exploring the use of mathematical models to design and optimize the layout of sports fields and other recreational spaces. This could involve using computer simulations to test different designs and identify the most efficient and effective layouts.

References

[1] "Mathematics for Engineers and Scientists" by Donald R. Hill
[2] "Calculus for Dummies" by Mark Ryan
[3] "Geometry for Dummies" by Mark Ryan
The High School Athletics Department's New Practice Field: A Q&A ===========================================================

Introduction

In our previous article, we explored the mathematical analysis of the high school athletics department's new practice field. In this article, we will answer some of the most frequently asked questions about the new practice field.

Q: What is the length and width of the new practice field?

A: The length of the new practice field is at least 40 meters, and the width is at least 25 meters.

Q: How was the length and width of the new practice field determined?

A: The length and width of the new practice field were determined by solving a system of equations. The system of equations was based on the area of the original field and the length and width of the new addition.

Q: What is the area of the new practice field?

A: The area of the new practice field is 2400 square meters.

Q: How does the new practice field compare to the original field?

A: The new practice field is larger than the original field. The original field had an area of 1200 square meters, while the new practice field has an area of 2400 square meters.

Q: What are the dimensions of the combined field?

A: The dimensions of the combined field are:

Length: 40 meters Width: 25 meters

Q: How was the width of the new addition determined?

A: The width of the new addition was determined by solving for w' in the equation (2w' + 4w - 10)(w' - 5) = 0. The solution to the equation was w' = 5.

Q: What are the real-world applications of the mathematical analysis of the new practice field?

A: The mathematical analysis of the new practice field has real-world applications in the field of engineering and architecture. The design of buildings, bridges, and other structures requires the use of mathematical equations to ensure that they are safe and functional.

Q: What are some potential future research areas in this field?

A: Some potential future research areas in this field include:

Exploring the use of mathematical models to design and optimize the layout of sports fields and other recreational spaces.
Using computer simulations to test different designs and identify the most efficient and effective layouts.
Investigating the use of mathematical equations to design and optimize the layout of other types of facilities, such as schools and hospitals.

Q: What are some of the benefits of using mathematical analysis in the design of sports fields and other recreational spaces?

A: Some of the benefits of using mathematical analysis in the design of sports fields and other recreational spaces include:

Ensuring that the facilities are safe and functional.
Optimizing the layout of the facilities to maximize space and efficiency.
Reducing costs by minimizing the amount of materials and labor required to build the facilities.

Conclusion

In conclusion, the high school athletics department's new practice field is a unique and interesting example of how mathematical analysis can be used to design and optimize the layout of sports fields and other recreational spaces. We hope that this Q&A article has provided you with a better understanding of the mathematical analysis of the new practice field and its real-world applications.