Finding the Volume of the Remaining Solid

Oct 13, 2025 by miodeya.com 111 views

Volume of the Remaining Solid: A Mathematical Exploration

In this article, we will delve into the world of mathematics and explore a problem that involves the removal of cones from a solid cylinder. The problem requires us to find the volume of the remaining solid after the cones have been scooped out. We will use the concepts of volume of a cylinder and volume of a cone to solve this problem.

We are given a solid cylinder with a height of 20 cm and a base radius of 7 cm. From each end of the cylinder, a cone with a base radius of 2.1 cm and a height of 5 cm is scooped out. Our goal is to find the volume of the remaining solid after the cones have been removed.

Volume of a Cylinder

The volume of a cylinder is given by the formula:

V = πr²h

where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cylinder.

In this case, the volume of the original cylinder is:

V = π(7)²(20) = 3.14 × 49 × 20 = 3084.8 cm³

Volume of a Cone

The volume of a cone is given by the formula:

V = (1/3)πr²h

where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cone.

In this case, the volume of each cone is:

V = (1/3)π(2.1)²(5) = (1/3) × 3.14 × 4.41 × 5 = 36.9 cm³

Finding the Volume of the Remaining Solid

To find the volume of the remaining solid, we need to subtract the volume of the two cones from the volume of the original cylinder. Since there are two cones, we need to multiply the volume of one cone by 2.

The volume of the remaining solid is:

V = 3084.8 - 2 × 36.9 = 3084.8 - 73.8 = 3011 cm³

In this article, we explored a problem that involved the removal of cones from a solid cylinder. We used the concepts of volume of a cylinder and volume of a cone to find the volume of the remaining solid. The volume of the remaining solid is 3011 cm³.

Volume of a cylinder: V = πr²h
Volume of a cone: V = (1/3)πr²h

Volume of a cylinder
Volume of a cone
Removal of cones from a solid cylinder

This problem has real-world applications in various fields such as engineering, architecture, and design. For example, in the design of a building, the volume of the remaining solid after the removal of a cone-shaped structure can be used to determine the amount of material needed for the construction.

This problem can be extended to more complex scenarios, such as the removal of multiple cones from a solid cylinder or the removal of a cone from a solid cylinder with a non-uniform cross-section. Further research can be conducted to explore these scenarios and develop more accurate mathematical models.

[1] "Mathematics for Engineers and Scientists" by Donald R. Hill
[2] "Calculus" by Michael Spivak
[3] "Geometry and Trigonometry" by I.M. Gelfand

Note: The references provided are for illustrative purposes only and are not actual references used in this article.
Frequently Asked Questions: Volume of the Remaining Solid

A: The volume of the original cylinder is 3084.8 cm³, which is calculated using the formula V = πr²h, where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cylinder.

A: The volume of each cone is 36.9 cm³, which is calculated using the formula V = (1/3)πr²h, where V is the volume, π is a constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cone.

A: To find the volume of the remaining solid, you need to subtract the volume of the two cones from the volume of the original cylinder. Since there are two cones, you need to multiply the volume of one cone by 2.

A: The volume of the remaining solid is 3011 cm³, which is calculated by subtracting the volume of the two cones from the volume of the original cylinder.

A: This problem has real-world applications in various fields such as engineering, architecture, and design. For example, in the design of a building, the volume of the remaining solid after the removal of a cone-shaped structure can be used to determine the amount of material needed for the construction.

A: Yes, this problem can be extended to more complex scenarios, such as the removal of multiple cones from a solid cylinder or the removal of a cone from a solid cylinder with a non-uniform cross-section. Further research can be conducted to explore these scenarios and develop more accurate mathematical models.

A: Some common mistakes to avoid when solving this problem include:

Not using the correct formula for the volume of a cylinder or cone
Not multiplying the volume of one cone by 2 when there are two cones
Not checking the units of the answer to ensure they are correct

A: You can practice solving this type of problem by working through example problems and checking your answers against the solutions. You can also try creating your own problems and solving them to see if you can come up with the correct answer.

A: Some resources for learning more about this topic include:

"Mathematics for Engineers and Scientists" by Donald R. Hill
"Calculus" by Michael Spivak
"Geometry and Trigonometry" by I.M. Gelfand

Note: The resources provided are for illustrative purposes only and are not actual resources used in this article.