Question
- Describe a physical model that you could use to illustrate the concept of a finite area
being enclosed by an unlimited perimeter.
Copyright © 2015 Pearson Education, Inc. All rights reserved.
being enclosed by an unlimited perimeter.
Copyright © 2015 Pearson Education, Inc. All rights reserved.
Ask by Dickson Griffiths. in the United States
Mar 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
You can use a circle to show a finite area with an unlimited perimeter. Draw a circle with a specific size. The area inside is limited, but if you make the circle bigger, the edge (perimeter) keeps getting longer without end.
Solution
To illustrate the concept of a finite area being enclosed by an unlimited perimeter, you can use the physical model of a circle with an infinitely thin boundary. Here’s how you can describe this model step by step:
-
Materials Needed:
- A large piece of paper or a whiteboard.
- A compass or a round object (like a cup) to trace a circle.
- A ruler to measure the radius.
-
Creating the Circle:
- Use the compass to draw a circle with a specific radius
. Alternatively, trace around a round object to create a circle. - Label the center of the circle as point
.
- Use the compass to draw a circle with a specific radius
-
Understanding the Area:
- The area
of the circle can be calculated using the formula: - This area is finite and represents the space enclosed within the circle.
- The area
-
Understanding the Perimeter:
- The perimeter of the circle, also known as the circumference
, can be calculated using the formula: - As the radius
increases, the circumference increases linearly, but it remains finite for any given radius.
- The perimeter of the circle, also known as the circumference
-
Illustrating the Concept of Unlimited Perimeter:
- To illustrate the concept of an unlimited perimeter, you can discuss the idea of a fractal boundary, such as the Koch snowflake.
- Start with an equilateral triangle and iteratively add smaller triangles to each side. As you continue this process infinitely, the perimeter increases without bound, while the area enclosed remains finite.
-
Conclusion:
- This model effectively demonstrates that while the area can be finite (like the area of a circle or the area of the Koch snowflake), the perimeter can become infinitely large through certain constructions or as the radius increases.
By using this physical model, you can visually and mathematically convey the concept of a finite area being enclosed by an unlimited perimeter.
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Simplify this solution Bonus Knowledge
Imagine a hula hoop! The hoop itself represents the unlimited perimeter, while the space it encloses represents the finite area. You can stretch or contort the hoop into different shapes, like an oval or even a funny figure like a star, yet the area inside will always remain a finite quantity, despite the potentially infinite length of the hoop’s edge.
Another fun model could be a soap bubble. When you blow a bubble, the surface creates an infinite perimeter, as the soap film is continuously thinner at the edges. However, the volume of air contained inside the bubble is limited, encapsulating a defined amount of space. This visual can help illustrate how an infinite length can enclose a finite area!
