Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Maximizing Area with Limited Fencing**
**Problem Description:**
You are tasked with fencing off two rectangular garden beds of the same size such that they share one side. Given that you have 120 meters of fencing, what is the maximum total area (in square meters) that you can fence off in this way?
**Diagram Explanation:**
The diagram shows two rectangles laid out side by side, sharing a common side. The length of the shared side is denoted as \( x \), and the width of each rectangle is denoted as \( y \).
The fencing requirements include:
- Two widths (\( 2y \))
- Two lengths (\( x \))
- One additional dividing length (\( x \))
The total length of the fencing used is given by the formula:
\[ 3x + 2y = 120 \]
The total area \( A \) of the two rectangles, which needs to be maximized, can be expressed as:
\[ A = 2xy \]
**Objective:**
Find the optimal values for \( x \) and \( y \) that maximize the total area \( A \) under the constraint of the available fencing.](https://content.bartleby.com/qna-images/question/d94ee947-1f00-4d5c-8d00-18458a43a16b/144f5d5c-9447-46c0-a2e1-0e587e7cf92e/0frck3m_processed.jpeg)
Transcribed Image Text:**Title: Maximizing Area with Limited Fencing**
**Problem Description:**
You are tasked with fencing off two rectangular garden beds of the same size such that they share one side. Given that you have 120 meters of fencing, what is the maximum total area (in square meters) that you can fence off in this way?
**Diagram Explanation:**
The diagram shows two rectangles laid out side by side, sharing a common side. The length of the shared side is denoted as \( x \), and the width of each rectangle is denoted as \( y \).
The fencing requirements include:
- Two widths (\( 2y \))
- Two lengths (\( x \))
- One additional dividing length (\( x \))
The total length of the fencing used is given by the formula:
\[ 3x + 2y = 120 \]
The total area \( A \) of the two rectangles, which needs to be maximized, can be expressed as:
\[ A = 2xy \]
**Objective:**
Find the optimal values for \( x \) and \( y \) that maximize the total area \( A \) under the constraint of the available fencing.
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