Juanita Has 3 Rectangular Cards That Are 7 Inches By 9 Inches. How Can She Arrange The Cards, Without Overlapping, To Make One Larger Polygon With The Smallest Possible Perimeter? How Will The Area Of The Polygon Compare To The Combined Area Of The 3
Introduction
Juanita has three rectangular cards, each measuring 7 inches by 9 inches. The challenge is to arrange these cards, without overlapping, to form a larger polygon with the smallest possible perimeter. In this article, we will delve into the mathematical concepts behind this problem and explore the optimal arrangement of the cards. We will also compare the area of the resulting polygon to the combined area of the three individual cards.
Understanding the Problem
To approach this problem, we need to consider the properties of the rectangular cards and the polygon they will form. Each card has a length of 9 inches and a width of 7 inches. The perimeter of a polygon is the sum of the lengths of its sides. To minimize the perimeter, we need to arrange the cards in a way that minimizes the total length of the sides.
Optimal Arrangement
The optimal arrangement of the cards is to form a rectangle with the cards placed side by side. This arrangement minimizes the perimeter because it eliminates the need for any additional sides. By placing the cards side by side, we create a rectangle with a length of 21 inches (3 x 7 inches) and a width of 9 inches.
Calculating the Perimeter
To calculate the perimeter of the resulting rectangle, we need to add up the lengths of all its sides. The perimeter is given by the formula:
Perimeter = 2(Length + Width)
Substituting the values, we get:
Perimeter = 2(21 + 9) Perimeter = 2 x 30 Perimeter = 60 inches
Comparing the Area
To compare the area of the resulting polygon to the combined area of the three individual cards, we need to calculate the area of the rectangle and the combined area of the cards.
Area of the Rectangle
The area of a rectangle is given by the formula:
Area = Length x Width
Substituting the values, we get:
Area = 21 x 9 Area = 189 square inches
Combined Area of the Cards
The combined area of the three individual cards is given by the formula:
Combined Area = 3 x (Length x Width)
Substituting the values, we get:
Combined Area = 3 x (7 x 9) Combined Area = 3 x 63 Combined Area = 189 square inches
Conclusion
In conclusion, the optimal arrangement of the cards is to form a rectangle with the cards placed side by side. This arrangement minimizes the perimeter because it eliminates the need for any additional sides. The perimeter of the resulting rectangle is 60 inches, and the area is 189 square inches. The combined area of the three individual cards is also 189 square inches, which means that the area of the resulting polygon is equal to the combined area of the cards.
Mathematical Concepts
This problem involves several mathematical concepts, including:
- Perimeter: The sum of the lengths of the sides of a polygon.
- Area: The amount of space inside a polygon.
- Rectangle: A type of polygon with four sides of equal length.
- Arrangement: The way in which objects are placed in relation to each other.
Real-World Applications
This problem has real-world applications in various fields, including:
- Architecture: Designing buildings and structures that minimize perimeter and maximize area.
- Engineering: Designing systems and mechanisms that minimize perimeter and maximize area.
- Art: Creating designs and patterns that minimize perimeter and maximize area.
Future Research Directions
Future research directions in this area could include:
- Optimizing perimeter and area for different shapes: Exploring the optimal arrangements of different shapes, such as triangles, quadrilaterals, and polygons.
- Minimizing perimeter and maximizing area for complex shapes: Investigating the optimal arrangements of complex shapes, such as fractals and self-similar patterns.
- Applying mathematical concepts to real-world problems: Developing mathematical models and algorithms to solve real-world problems in fields such as architecture, engineering, and art.
Juanita's Polygon Puzzle: A Mathematical Exploration - Q&A =====================================================
Introduction
In our previous article, we explored the mathematical concepts behind Juanita's polygon puzzle, where she had to arrange three rectangular cards to form a larger polygon with the smallest possible perimeter. We also compared the area of the resulting polygon to the combined area of the three individual cards. In this article, we will answer some frequently asked questions related to this problem.
Q&A
Q: What is the optimal arrangement of the cards to form a polygon with the smallest possible perimeter?
A: The optimal arrangement of the cards is to form a rectangle with the cards placed side by side. This arrangement minimizes the perimeter because it eliminates the need for any additional sides.
Q: How do you calculate the perimeter of the resulting polygon?
A: To calculate the perimeter of the resulting polygon, you need to add up the lengths of all its sides. The perimeter is given by the formula:
Perimeter = 2(Length + Width)
Q: What is the area of the resulting polygon?
A: The area of the resulting polygon is given by the formula:
Area = Length x Width
Q: How does the area of the resulting polygon compare to the combined area of the three individual cards?
A: The area of the resulting polygon is equal to the combined area of the three individual cards, which is 189 square inches.
Q: What are some real-world applications of this problem?
A: This problem has real-world applications in various fields, including architecture, engineering, and art. Designers and engineers use mathematical concepts like perimeter and area to create efficient and effective designs.
Q: Can you apply this problem to other shapes, such as triangles or quadrilaterals?
A: Yes, you can apply this problem to other shapes, such as triangles or quadrilaterals. However, the optimal arrangement of the cards will depend on the specific shape and its properties.
Q: How can you minimize the perimeter and maximize the area of a polygon?
A: To minimize the perimeter and maximize the area of a polygon, you need to arrange the cards in a way that minimizes the total length of the sides while maximizing the area. This can be achieved by using mathematical concepts like perimeter and area, and by experimenting with different arrangements of the cards.
Mathematical Concepts
This problem involves several mathematical concepts, including:
- Perimeter: The sum of the lengths of the sides of a polygon.
- Area: The amount of space inside a polygon.
- Rectangle: A type of polygon with four sides of equal length.
- Arrangement: The way in which objects are placed in relation to each other.
Real-World Applications
This problem has real-world applications in various fields, including:
- Architecture: Designing buildings and structures that minimize perimeter and maximize area.
- Engineering: Designing systems and mechanisms that minimize perimeter and maximize area.
- Art: Creating designs and patterns that minimize perimeter and maximize area.
Future Research Directions
Future research directions in this area could include:
- Optimizing perimeter and area for different shapes: Exploring the optimal arrangements of different shapes, such as triangles, quadrilaterals, and polygons.
- Minimizing perimeter and maximizing area for complex shapes: Investigating the optimal arrangements of complex shapes, such as fractals and self-similar patterns.
- Applying mathematical concepts to real-world problems: Developing mathematical models and algorithms to solve real-world problems in fields such as architecture, engineering, and art.
Conclusion
In conclusion, Juanita's polygon puzzle is a classic problem that involves mathematical concepts like perimeter and area. By understanding the optimal arrangement of the cards and the properties of the resulting polygon, we can apply this problem to real-world scenarios and develop new mathematical models and algorithms.