The area of a triangle with sides of length a, b, and c is Js(s – a)(s – b)(s – c), where s is half the perimeter of the triangle. We have 60 feet of fence and want to fence a triangular-shaped area. Part A: Formulate the problem as a constrained nonlinear program that will enable us to maximize the are of the fenced area, with constraints. Clearly indicate the variables, objective function, and constraints. Hint: The length of a side of a triangle must be less than or equal to the sum of the lengths of the other two sides. Part B: Solve the Program (provide exact values for all variables and the optimal objective function).
The area of a triangle with sides of length a, b, and c is Js(s – a)(s – b)(s – c), where s is half the perimeter of the triangle. We have 60 feet of fence and want to fence a triangular-shaped area. Part A: Formulate the problem as a constrained nonlinear program that will enable us to maximize the are of the fenced area, with constraints. Clearly indicate the variables, objective function, and constraints. Hint: The length of a side of a triangle must be less than or equal to the sum of the lengths of the other two sides. Part B: Solve the Program (provide exact values for all variables and the optimal objective function).
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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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57347d55-1fa8-4eea-a8a6-6e37e3b644e7%2Fc89fe962-3ea9-4574-b81c-0458f88dc3b4%2Foev9ssp_processed.png&w=3840&q=75)
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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