The Surface Area Of A 30-inch Foam Exercise roller Is 19877 Square Inches, And The Rolling surface Has An Area Of 18077 Square Inches. What Is The Circumference Of The Roller? Express Your Answer In Terms Of ΠT. a

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Introduction

In this article, we will delve into the world of mathematics, specifically geometry, to solve a problem related to the surface area of a foam exercise roller. The problem states that the surface area of a 30-inch foam exercise roller is 19877 square inches, and the rolling surface has an area of 18077 square inches. Our goal is to find the circumference of the roller in terms of πT.

Understanding the Problem

To approach this problem, we need to understand the concept of surface area and how it relates to the circumference of a circle. The surface area of a circle is given by the formula:

A = πr^2

where A is the surface area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Breaking Down the Problem

Let's break down the problem into smaller parts. We are given two pieces of information:

  1. The surface area of the foam exercise roller is 19877 square inches.
  2. The rolling surface has an area of 18077 square inches.

We can use these two pieces of information to find the circumference of the roller.

Step 1: Finding the Radius of the Roller

First, let's find the radius of the roller. We know that the surface area of the roller is 19877 square inches, and the rolling surface has an area of 18077 square inches. The difference between these two areas is the area of the top and bottom of the roller, which is given by:

A_top_bottom = A_roller - A_rolling

Substituting the given values, we get:

A_top_bottom = 19877 - 18077 A_top_bottom = 1800

The area of the top and bottom of the roller is 1800 square inches. Since the top and bottom are circular, we can use the formula for the area of a circle to find the radius:

A_top_bottom = πr^2

Substituting the value of A_top_bottom, we get:

1800 = πr^2

Solving for r, we get:

r = √(1800/π) r ≈ 17.32 inches

Step 2: Finding the Circumference of the Roller

Now that we have the radius of the roller, we can find the circumference. The circumference of a circle is given by the formula:

C = 2πr

Substituting the value of r, we get:

C = 2π(17.32) C ≈ 109.04π inches

Conclusion

In this article, we used the surface area of a foam exercise roller to find the circumference of the roller in terms of πT. We broke down the problem into smaller parts, finding the radius of the roller and then using it to find the circumference. The final answer is approximately 109.04π inches.

Mathematical Formulation

The surface area of the roller is given by:

A_roller = πr^2 + 2πr^2 A_roller = 3πr^2

The rolling surface has an area of:

A_rolling = πr^2

The difference between these two areas is the area of the top and bottom of the roller:

A_top_bottom = A_roller - A_rolling A_top_bottom = 2πr^2

The area of the top and bottom of the roller is 1800 square inches. Since the top and bottom are circular, we can use the formula for the area of a circle to find the radius:

A_top_bottom = πr^2

Substituting the value of A_top_bottom, we get:

1800 = πr^2

Solving for r, we get:

r = √(1800/π) r ≈ 17.32 inches

The circumference of the roller is given by:

C = 2πr C = 2π(17.32) C ≈ 109.04π inches

References

Discussion

This problem is a great example of how mathematics can be used to solve real-world problems. The surface area of a foam exercise roller may seem like a trivial problem, but it requires a deep understanding of geometry and mathematical formulas. The solution to this problem can be applied to a variety of real-world situations, such as designing exercise equipment or calculating the surface area of a sphere.

Conclusion

Introduction

In our previous article, we explored the surface area of a foam exercise roller and used it to find the circumference of the roller in terms of πT. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the surface area of a foam exercise roller?

A: The surface area of a foam exercise roller is given by the formula:

A_roller = πr^2 + 2πr^2 A_roller = 3πr^2

Q: How do I find the radius of the roller?

A: To find the radius of the roller, you need to know the surface area of the roller and the area of the rolling surface. The difference between these two areas is the area of the top and bottom of the roller, which is given by:

A_top_bottom = A_roller - A_rolling A_top_bottom = 2πr^2

You can then use the formula for the area of a circle to find the radius:

A_top_bottom = πr^2

Substituting the value of A_top_bottom, you get:

1800 = πr^2

Solving for r, you get:

r = √(1800/π) r ≈ 17.32 inches

Q: How do I find the circumference of the roller?

A: Once you have the radius of the roller, you can find the circumference using the formula:

C = 2πr C = 2π(17.32) C ≈ 109.04π inches

Q: What is the significance of π in the formula for circumference?

A: π (pi) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14. In the formula for circumference, π is used to calculate the circumference of the roller in terms of its radius.

Q: Can I use this formula to find the circumference of any circle?

A: Yes, you can use this formula to find the circumference of any circle, as long as you know the radius of the circle. The formula is:

C = 2πr

Q: What are some real-world applications of this formula?

A: This formula has many real-world applications, such as:

  • Designing exercise equipment, like foam rollers or exercise balls
  • Calculating the surface area of a sphere or a cylinder
  • Finding the circumference of a circle in engineering or architecture

Q: Can I use a calculator to find the circumference of the roller?

A: Yes, you can use a calculator to find the circumference of the roller. Simply enter the value of the radius and the value of π, and the calculator will give you the circumference.

Conclusion

In conclusion, the surface area of a foam exercise roller can be used to find the circumference of the roller in terms of πT. By answering some of the most frequently asked questions related to this topic, we hope to have provided a better understanding of this concept.