Maximizing Area of a Rectangle Inside a Circle: A Comprehensive Guide

Maximizing Area of a Rectangle Inside a Circle: A Comprehensive Guide 📐🔬💻📊📈️⚛️🧩🎯

Introduction

In this tutorial, we'll be solving an interesting optimization problem - finding the maximum area of a rectangle inside a circle.

We have a circle with radiusr, and our goal is to determine the largest possible rectangle that can fit inside this circle. Let's dive in!

Formulating the Problem Mathematically

Let's denote the length of the rectangle asx(on the x-axis) and its width asy(on the y-axis). The area of the rectangle will then be 4 times the product ofxandy. However, we have some constraints to consider:

1. x must be between -r and r.
2. y must also be between -r and r.
3. Additionally, the equation x^2 + y^2 = r^2 holds true since both points belong to the circle.

Solving the Problem Using Python

We'll be using Pyomo, a popular open-source Python package for mathematical modeling and optimization, to solve this problem.

First, we import the necessary modules and define some input data and decision variables:
```python
from pyomo.environ import *

model = ConcreteModel()

R = Param(initial=10)
X = Var(bounds=(0, R))
Y = Var(bounds=(0, R))

Constraints

model.c1 = ConstraintList()
model.c1.add(model.x2 + model.y2 == R**2)

Objective function

model.obj = Objective(rule=4 * model.x * model.y, sense=maximize)
```

Solving the Optimization Problem

We'll use a non-linear solver to solve our problem due to its multiplicative nature. Here, we choose IPOPT:
```python
from pyomo.opt import SolverFactory

solver = SolverFactory('ipopt')
results = solver(model)
```

Analyzing the Results

Now that we have solved our problem, let's examine the results:python
x_val = model.x()
y_val = model.y()
area = 4 * x_val * y_val
print(f'Maximum area: {area}')

Conclusion and Further Exploration

We have successfully solved the optimization problem of finding the maximum area of a rectangle inside a circle using Python and Pyomo. With the right mathematical formulation and an appropriate solver, we were able to find the optimal solution!

For those interested in further exploration, you can modify this model for various applications or even tackle more complex problems using similar techniques.

Stay tuned for more insights on optimization, mathematics, and programming! 📚