A Sculptor Created A Design By Carving A Cone Out Of A Cylinder. The Cone And Cylinder Share The Same Radius And Height. If The Volume Of The Cylinder Before Removing The Cone Is $54 \, \text{in}^3$, What Is The Volume Of The Amount

Introduction

In the world of mathematics, shapes and their properties are the foundation of various concepts and theories. One such concept is the relationship between a cylinder and a cone, where the cone is carved out of the cylinder. In this article, we will delve into the world of geometry and explore the volume of the remaining cylinder after the cone is removed. We will use the given information that the volume of the cylinder before removing the cone is $54 \, \text{in}^3$ and determine the volume of the remaining amount.

The Volume of a Cylinder

Before we dive into the problem, let's recall the formula for the volume of a cylinder. The volume of a cylinder is given by the formula:

$$V = \pi r^2 h$$

where $r$ is the radius of the cylinder and $h$ is its height.

The Volume of a Cone

The volume of a cone is given by the formula:

$$V = \frac{1}{3} \pi r^2 h$$

where $r$ is the radius of the cone and $h$ is its height.

The Relationship Between the Cylinder and the Cone

Since the cone is carved out of the cylinder, they share the same radius and height. Let's denote the radius and height of both the cylinder and the cone as $r$ and $h$, respectively.

The Volume of the Remaining Cylinder

The volume of the remaining cylinder after the cone is removed is equal to the volume of the original cylinder minus the volume of the cone. Using the formulas for the volume of a cylinder and a cone, we can write:

$$V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}}$$

Substituting the formulas for the volume of a cylinder and a cone, we get:

$$V_{\text{remaining}} = \pi r^2 h - \frac{1}{3} \pi r^2 h$$

Simplifying the expression, we get:

$$V_{\text{remaining}} = \frac{2}{3} \pi r^2 h$$

Given Information

We are given that the volume of the cylinder before removing the cone is $54 \, \text{in}^3$. Using the formula for the volume of a cylinder, we can write:

$$\pi r^2 h = 54$$

Solving for the Volume of the Remaining Cylinder

Now that we have the equation for the volume of the remaining cylinder, we can substitute the given information to solve for the volume. Rearranging the equation, we get:

$$V_{\text{remaining}} = \frac{2}{3} \pi r^2 h = \frac{2}{3} \cdot 54$$

Simplifying the expression, we get:

$$V_{\text{remaining}} = 36$$

Therefore, the volume of the remaining cylinder after the cone is removed is $36 \, \text{in}^3$.

Conclusion

In this article, we explored the volume of a carved cylinder and determined the volume of the remaining amount after the cone is removed. We used the given information that the volume of the cylinder before removing the cone is $54 \, \text{in}^3$ and applied the formulas for the volume of a cylinder and a cone to solve for the volume of the remaining cylinder. The result is a volume of $36 \, \text{in}^3$, which is a testament to the power of mathematical reasoning and problem-solving.

References

• [1] "Geometry: A Comprehensive Introduction" by Michael Spivak
• [2] "Calculus: Early Transcendentals" by James Stewart

Additional Resources

• [1] Khan Academy: Geometry and Trigonometry
• [2] MIT OpenCourseWare: Calculus and Linear Algebra
  A Sculptor's Masterpiece: Unveiling the Volume of a Carved Cylinder ===========================================================

Q&A: Frequently Asked Questions

In this section, we will address some of the most frequently asked questions related to the volume of a carved cylinder.

Q: What is the formula for the volume of a cylinder?

A: The formula for the volume of a cylinder is:

$$V = \pi r^2 h$$

where $r$ is the radius of the cylinder and $h$ is its height.

Q: What is the formula for the volume of a cone?

A: The formula for the volume of a cone is:

$$V = \frac{1}{3} \pi r^2 h$$

where $r$ is the radius of the cone and $h$ is its height.

Q: How do I calculate the volume of the remaining cylinder after the cone is removed?

A: To calculate the volume of the remaining cylinder after the cone is removed, you need to subtract the volume of the cone from the volume of the original cylinder. The formula for the volume of the remaining cylinder is:

$$V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}}$$

Q: What is the relationship between the cylinder and the cone?

A: Since the cone is carved out of the cylinder, they share the same radius and height. Let's denote the radius and height of both the cylinder and the cone as $r$ and $h$, respectively.

Q: How do I use the given information to solve for the volume of the remaining cylinder?

A: We are given that the volume of the cylinder before removing the cone is $54 \, \text{in}^3$. Using the formula for the volume of a cylinder, we can write:

$$\pi r^2 h = 54$$

To solve for the volume of the remaining cylinder, we need to substitute this equation into the formula for the volume of the remaining cylinder:

$$V_{\text{remaining}} = \frac{2}{3} \pi r^2 h = \frac{2}{3} \cdot 54$$

Simplifying the expression, we get:

$$V_{\text{remaining}} = 36$$

Q: What is the volume of the remaining cylinder after the cone is removed?

A: The volume of the remaining cylinder after the cone is removed is $36 \, \text{in}^3$.

Q: Can I use this formula to calculate the volume of any cylinder and cone?

A: Yes, you can use this formula to calculate the volume of any cylinder and cone, as long as they share the same radius and height.

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

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

• Calculating the volume of a container or a tank
• Determining the volume of a material or a substance
• Designing and building structures, such as bridges or buildings
• Understanding the behavior of fluids and gases

Conclusion

In this article, we addressed some of the most frequently asked questions related to the volume of a carved cylinder. We provided formulas and explanations for calculating the volume of a cylinder and a cone, as well as the volume of the remaining cylinder after the cone is removed. We also discussed some real-world applications of this concept and provided additional resources for further learning.

References

• [1] "Geometry: A Comprehensive Introduction" by Michael Spivak
• [2] "Calculus: Early Transcendentals" by James Stewart

Additional Resources

• [1] Khan Academy: Geometry and Trigonometry
• [2] MIT OpenCourseWare: Calculus and Linear Algebra