Finding the Possible Circles

Oct 13, 2025 by leolavanderia.com.br 62 views

Introduction

In the world of geometry, circles are a fundamental shape that can be found in various objects, including vases. Liliana, a crafty individual, is making a vase with a circular base. She wants the area of the base to be between 135 cm^2 and 155 cm^2. In this article, we will explore which circle could represent the base of the vase, using the value of pi (π) as 3.14.

Understanding the Problem

To find the perfect circle for Liliana's vase, we need to understand the relationship between the area of a circle and its radius. The formula for the area of a circle is given by:

A = πr^2

where A is the area and r is the radius of the circle.

Calculating the Radius

We are given that the area of the base should be between 135 cm^2 and 155 cm^2. We can use this information to find the range of possible radii for the circle.

First, let's find the minimum radius (r_min) that would give an area of 135 cm^2:

135 = πr_min^2 135 = 3.14 × r_min^2 r_min^2 = 135 / 3.14 r_min^2 = 43.03 r_min = √43.03 r_min ≈ 6.56 cm

Next, let's find the maximum radius (r_max) that would give an area of 155 cm^2:

155 = πr_max^2 155 = 3.14 × r_max^2 r_max^2 = 155 / 3.14 r_max^2 = 49.36 r_max = √49.36 r_max ≈ 7.02 cm

Finding the Possible Circles

Now that we have the range of possible radii (6.56 cm ≤ r ≤ 7.02 cm), we can find the possible circles that could represent the base of the vase.

The smallest possible circle would have a radius of 6.56 cm, and the largest possible circle would have a radius of 7.02 cm.

Conclusion

In conclusion, the possible circles that could represent the base of Liliana's vase are those with radii between 6.56 cm and 7.02 cm. This range of radii would give an area of the base between 135 cm^2 and 155 cm^2, satisfying Liliana's requirements.

Key Takeaways

The area of a circle is given by the formula A = πr^2.
To find the range of possible radii for a circle, we can use the given area range and the formula for the area of a circle.
The smallest possible circle would have a radius of 6.56 cm, and the largest possible circle would have a radius of 7.02 cm.

Real-World Applications

This problem has real-world applications in various fields, such as:

Crafting: Liliana's vase is a great example of how math is used in crafting. Understanding the relationship between the area of a circle and its radius is essential in creating objects with specific dimensions.
Engineering: In engineering, the area of a circle is used to calculate the volume of objects, such as pipes and tubes.
Architecture: Architects use the area of a circle to design buildings and structures with specific dimensions.

Final Thoughts

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is given by:

A = πr^2

where A is the area and r is the radius of the circle.

Q: Why is it important to find the range of possible radii for the circle?

A: Finding the range of possible radii for the circle is essential in determining the possible circles that could represent the base of Liliana's vase. This range of radii would give an area of the base between 135 cm^2 and 155 cm^2, satisfying Liliana's requirements.

Q: How do I calculate the minimum and maximum radii for the circle?

A: To calculate the minimum and maximum radii for the circle, you can use the given area range and the formula for the area of a circle. For example, if the area of the base should be between 135 cm^2 and 155 cm^2, you can find the minimum radius (r_min) that would give an area of 135 cm^2 and the maximum radius (r_max) that would give an area of 155 cm^2.

Q: What is the smallest possible circle that could represent the base of Liliana's vase?

A: The smallest possible circle that could represent the base of Liliana's vase would have a radius of 6.56 cm.

Q: What is the largest possible circle that could represent the base of Liliana's vase?

A: The largest possible circle that could represent the base of Liliana's vase would have a radius of 7.02 cm.

Q: How do I use the range of possible radii to find the possible circles that could represent the base of Liliana's vase?

A: To find the possible circles that could represent the base of Liliana's vase, you can use the range of possible radii (6.56 cm ≤ r ≤ 7.02 cm) to determine the possible circles that would give an area of the base between 135 cm^2 and 155 cm^2.

Q: What are some real-world applications of finding the perfect circle for Liliana's vase?

A: This problem has real-world applications in various fields, such as:

Crafting: Liliana's vase is a great example of how math is used in crafting. Understanding the relationship between the area of a circle and its radius is essential in creating objects with specific dimensions.
Engineering: In engineering, the area of a circle is used to calculate the volume of objects, such as pipes and tubes.
Architecture: Architects use the area of a circle to design buildings and structures with specific dimensions.

Q: What is the value of pi (π) used in this problem?

A: The value of pi (π) used in this problem is 3.14.

Q: How can I apply the concepts learned in this problem to other real-world situations?

A: The concepts learned in this problem can be applied to other real-world situations where the area of a circle is used to calculate the volume of objects or to design buildings and structures with specific dimensions.