Circle Equation Problem-Based Learning Scenario

Problem Overview

In this problem-based learning scenario, students will explore the equation of a circle through the lens of designing an optimal wireless charging system for a modern smart classroom. Students will apply their understanding of circle equations, coordinate geometry, and optimization to determine the best placement of wireless charging transmitters to ensure all student devices can receive power efficiently.

Learning Objectives

• Understand and apply the standard form of a circle equation: (x-h)²+(y-k)²=r²

• Convert between standard and general forms of circle equations: x² + y² + Dₓ + Eᵧ + F = 0

• Determine the center and radius of a circle from its equation

• Calculate distances between points in a coordinate plane

• Apply circle properties to solve real-world problems

• Optimize placement of objects based on geometric constraints

Real-World Context

Modern classrooms are increasingly reliant on technology, with students using tablets, laptops, and other devices. Wireless charging technology is advancing rapidly, allowing devices to charge without physical cables when within a certain radius of a charging transmitter. However, these transmitters have limited range and can be expensive to install.

Your school is planning to upgrade to a "smart classroom" with wireless charging capabilities. The administration wants to ensure that all student desks can receive wireless charging while minimizing the number of transmitters needed to reduce costs.

Problem Statement

Your team has been tasked with designing the optimal wireless charging system for your new smart classroom. The classroom can be represented on a coordinate grid where each unit represents 1 meter. Student desks are positioned at various coordinates throughout the room.

Each wireless charging transmitter creates a circular charging field with a radius of 3 meters. Any device within this circle can receive power. Your goal is to determine the minimum number of transmitters needed and their optimal placement to ensure all student desks are within range of at least one transmitter.

Classroom Specifications:

• The classroom dimensions are 10m × 8m (represented as a 10 × 8 coordinate grid)

• There are 24 student desks at fixed positions (use the modifier tool to get the coordinates)

• Each transmitter costs 25,000 Ft and creates a circular charging field with a 3m radius

• The budget allows for a maximum of 4 transmitters

Guided Questions

Initial Exploration

1. How can we represent a wireless charging transmitter's range mathematically?

2. If a transmitter is placed at position (4, 3), write the equation of the circle that represents its charging range.

3. For a desk at position (7, 5), determine if it falls within the range of the transmitter at (4, 3).

Developing a Strategy

1. What information do we need to determine if a point (desk) lies inside, outside, or on a circle?

2. How can we test multiple potential transmitter locations efficiently?

3. What mathematical approach could help us find optimal transmitter placement?

Implementation

1. If we place transmitters at positions (3, 2) and (7, 6), write the equations of both charging fields.

2. Which student desks are covered by these transmitters? Which remain uncovered?

3. How might we determine the best position for a third transmitter to cover the remaining desks?

Optimization

1. What is the minimum number of transmitters needed to cover all desks?

2. How can we verify that our solution is optimal?

3. If we could move some desks, how might we reduce the number of required transmitters?

Expected Solution Path

Approach 1: Geometric Visualization and Testing

1. Understand the Problem: Students plot all desk positions on a coordinate grid.

2. Represent Charging Fields: Students recognize that each transmitter creates a circle with equation (x-h)² + (y-k)² = 9, where (h,k) is the transmitter location and the radius is 3m.

3. Test Strategic Locations: Students identify clusters of desks and test transmitter placements.

4. Verify Coverage: For each potential transmitter location (h,k), students check if each desk position (x,y) satisfies (x-h)² + (y-k)² ≤ 9.

5. Optimize: Students iteratively adjust transmitter positions to minimize the number needed.

Sample Solution:

After testing various configurations, students might determine that three transmitters at positions (2.5, 2), (7.5, 2), and (5, 6) can cover all 24 desks. They verify this by showing that each desk position (xᵢ, yᵢ) satisfies at least one of these inequalities:

• (xᵢ - 2.5)² + (yᵢ - 2)² ≤ 9

• (xᵢ - 7.5)² + (yᵢ - 2)² ≤ 9

• (xᵢ - 5)² + (yᵢ - 6)² ≤ 9

Approach 2: Analytical Optimization

1. Formulate Mathematically: Students define the problem as finding the minimum number of circles with radius 3 that cover all given points.

2. Apply Circle Covering Algorithm: Students research and apply algorithms for the circle covering problem.

3. Use Center-Point Method: Students identify potential circle centers by finding the midpoint between pairs of desk positions.

4. Test Coverage: For each potential center, students determine how many desks fall within the 3m radius.

5. Select Optimal Centers: Students select the minimum number of centers that collectively cover all desks.

Sample Solution:

Students might use a greedy algorithm approach:

1. Place the first transmitter to cover the maximum number of desks

2. Remove covered desks from consideration

3. Repeat until all desks are covered

This might yield a solution with three transmitters at positions (3, 2.5), (8, 3), and (5, 6.5).

Approach 3: Computational Simulation

1. Create a Mathematical Model: Students develop a function that calculates coverage for any transmitter configuration.

2. Generate Test Cases: Students systematically test different transmitter locations.

3. Apply Optimization Techniques: Students might use concepts from calculus or computational methods to find optimal positions.

4. Validate Results: Students verify their solution meets all constraints and is cost-effective.

Sample Solution:

Students might write a program or use a spreadsheet to simulate different configurations, finding that three transmitters at positions (2, 2), (8, 2), and (5, 6) provide optimal coverage.

Extension Opportunities

1. Budget Optimization: If each transmitter costs 25,000 Ft, and moving a desk costs 5,000 Ft, would it be more cost-effective to rearrange some desks to reduce the number of required transmitters?

2. Power Efficiency: If the charging efficiency decreases with distance from the transmitter according to the formula E = 100 - 10d² (where d is the distance in meters and E is the percentage efficiency), determine the average charging efficiency for your configuration.

3. 3D Extension: If the transmitters are mounted at a height of 2m from the floor, and the desks are 0.75m high, how does this affect the charging radius on the desk surface? (This introduces the concept of 3D distance and spherical equations.)

4. Dynamic Classroom: Design an algorithm that could automatically recalculate optimal transmitter positions if desk arrangements change throughout the school year.

5. Real Data Collection: Measure your actual classroom and desk positions, then apply your mathematical model to determine the optimal wireless charging solution for your specific classroom.