Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Maximizing the Area of a Triangular Fence**
**Introduction:**
The area of a triangle with sides of length \(a\), \(b\), and \(c\) is given by the formula:
\[
\sqrt{s(s-a)(s-b)(s-c)}
\]
where \(s\) is half the perimeter of the triangle.
**Problem Statement:**
We have 60 feet of fence available and aim to enclose a triangular-shaped area.
**Part A: Formulating the Problem**
We need to set up a constrained nonlinear program to maximize the area of the fenced triangular region. The task requires us to define:
1. **Variables:**
- \(a\), \(b\), and \(c\): lengths of the sides of the triangle.
2. **Objective Function:**
- Maximize the area: \(\sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \frac{a+b+c}{2}\).
3. **Constraints:**
- The total length of the fence is 60 feet: \(a + b + c = 60\).
- Triangle inequality constraints:
- \(a \leq b + c\)
- \(b \leq a + c\)
- \(c \leq a + b\)
**Hint:**
The length of any side of a triangle must be less than or equal to the sum of the lengths of the other two sides.
**Part B: Solving the Program**
Calculate the exact values for all variables \(a\), \(b\), \(c\), and find the optimal value of the objective function (maximized area).
**Conclusion:**
This exercise requires an understanding of both geometry and optimization techniques to derive the maximum possible fenced area, respecting all given constraints.](https://content.bartleby.com/qna-images/question/57347d55-1fa8-4eea-a8a6-6e37e3b644e7/c89fe962-3ea9-4574-b81c-0458f88dc3b4/oev9ssp_processed.png)
Transcribed Image Text:**Title: Maximizing the Area of a Triangular Fence**
**Introduction:**
The area of a triangle with sides of length \(a\), \(b\), and \(c\) is given by the formula:
\[
\sqrt{s(s-a)(s-b)(s-c)}
\]
where \(s\) is half the perimeter of the triangle.
**Problem Statement:**
We have 60 feet of fence available and aim to enclose a triangular-shaped area.
**Part A: Formulating the Problem**
We need to set up a constrained nonlinear program to maximize the area of the fenced triangular region. The task requires us to define:
1. **Variables:**
- \(a\), \(b\), and \(c\): lengths of the sides of the triangle.
2. **Objective Function:**
- Maximize the area: \(\sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \frac{a+b+c}{2}\).
3. **Constraints:**
- The total length of the fence is 60 feet: \(a + b + c = 60\).
- Triangle inequality constraints:
- \(a \leq b + c\)
- \(b \leq a + c\)
- \(c \leq a + b\)
**Hint:**
The length of any side of a triangle must be less than or equal to the sum of the lengths of the other two sides.
**Part B: Solving the Program**
Calculate the exact values for all variables \(a\), \(b\), \(c\), and find the optimal value of the objective function (maximized area).
**Conclusion:**
This exercise requires an understanding of both geometry and optimization techniques to derive the maximum possible fenced area, respecting all given constraints.
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