A school wishes to enclose its rectangular playground using 120 meters of fencing. Suppose that a side length (in meters) of the playground is x, as shown below. (a) Find a function that gives the area A (x) of the playground (in square meters) in 0
A school wishes to enclose its rectangular playground using 120 meters of fencing. Suppose that a side length (in meters) of the playground is x, as shown below. (a) Find a function that gives the area A (x) of the playground (in square meters) in 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
A school wishes to enclose its rectangular playground using 120 meters of fencing. Suppose that a side length (in meters) of the playground is \( x \), as shown below.
[Diagram of the rectangular playground with side length \( x \).]
### Questions
(a) **Find a function that gives the area \( A(x) \) of the playground (in square meters) in terms of \( x \).**
\[ A(x) = \]
(b) **What side length \( x \) gives the maximum area that the playground can have?**
\[ \text{Side length } x = \, \] meters
(c) **What is the maximum area that the playground can have?**
\[ \text{Maximum area: } \] square meters
### Explanation of Diagram
The diagram illustrates a rectangular playground where one side length is denoted as \( x \). The rectangular shape is constructed with a total perimeter of 120 meters of fencing.
To find the function for the area \( A(x) \), understand that the perimeter \( P \) of a rectangle is given by:
\[ P = 2x + 2y \]
where \( x \) and \( y \) are the length and the width of the rectangle, respectively. Given that the perimeter is 120 meters, we have:
\[ 2x + 2y = 120 \]
Solving for \( y \):
\[ y = \frac{120 - 2x}{2} \]
The area \( A \) of the rectangle is given by \( A(x) = x \cdot y \). Substituting the value of \( y \) from above:
\[ A(x) = x \left( \frac{120 - 2x}{2} \right) = x \left( 60 - x \right) \]
Thus, the function:
\[ A(x) = 60x - x^2 \]
### Notes for Educators
* Explain the relationship between perimeter and area for rectangles.
* Guide students through algebraic manipulation to solve for one variable in terms of the other.
* Discuss optimizing the area by completing the square or using derivative techniques in calculus to find maximum values.
This structured approach helps familiarize students with practical applications of algebra and geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbae73a37-719e-4892-8745-1f816a181ea7%2Fba968284-6d66-4135-a14c-ddd898bb6882%2Ft0cdfga_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A school wishes to enclose its rectangular playground using 120 meters of fencing. Suppose that a side length (in meters) of the playground is \( x \), as shown below.
[Diagram of the rectangular playground with side length \( x \).]
### Questions
(a) **Find a function that gives the area \( A(x) \) of the playground (in square meters) in terms of \( x \).**
\[ A(x) = \]
(b) **What side length \( x \) gives the maximum area that the playground can have?**
\[ \text{Side length } x = \, \] meters
(c) **What is the maximum area that the playground can have?**
\[ \text{Maximum area: } \] square meters
### Explanation of Diagram
The diagram illustrates a rectangular playground where one side length is denoted as \( x \). The rectangular shape is constructed with a total perimeter of 120 meters of fencing.
To find the function for the area \( A(x) \), understand that the perimeter \( P \) of a rectangle is given by:
\[ P = 2x + 2y \]
where \( x \) and \( y \) are the length and the width of the rectangle, respectively. Given that the perimeter is 120 meters, we have:
\[ 2x + 2y = 120 \]
Solving for \( y \):
\[ y = \frac{120 - 2x}{2} \]
The area \( A \) of the rectangle is given by \( A(x) = x \cdot y \). Substituting the value of \( y \) from above:
\[ A(x) = x \left( \frac{120 - 2x}{2} \right) = x \left( 60 - x \right) \]
Thus, the function:
\[ A(x) = 60x - x^2 \]
### Notes for Educators
* Explain the relationship between perimeter and area for rectangles.
* Guide students through algebraic manipulation to solve for one variable in terms of the other.
* Discuss optimizing the area by completing the square or using derivative techniques in calculus to find maximum values.
This structured approach helps familiarize students with practical applications of algebra and geometry.
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