Calculating the Radius of the Conical Part

Oct 13, 2025 by talynt.org 20 views

Introduction

In this article, we will delve into the world of geometry and explore the properties of a specific toy that consists of a cone mounted on a cylinder. The toy's dimensions are provided, and we will use this information to calculate various geometric parameters. This discussion will not only help us understand the geometry of the toy but also provide a practical application of mathematical concepts.

Given Information

The total height of the toy is 110 mm.
The height of the conical part is 77 mm.
The diameter of the base of the conical part is 72 mm.
The diameter of the cylindrical part is 40 mm.

Calculating the Radius of the Conical Part

To calculate the radius of the conical part, we need to divide the diameter by 2.

Radius of the conical part = 72 mm / 2 = 36 mm

Calculating the Radius of the Cylindrical Part

Similarly, we can calculate the radius of the cylindrical part by dividing its diameter by 2.

Radius of the cylindrical part = 40 mm / 2 = 20 mm

Calculating the Slant Height of the Conical Part

The slant height of the conical part can be calculated using the Pythagorean theorem. We know the height of the conical part (77 mm) and the radius of its base (36 mm). We can use these values to find the slant height.

Slant height of the conical part = √(height^2 + radius^2)
Slant height of the conical part = √(77^2 + 36^2)
Slant height of the conical part = √(5929 + 1296)
Slant height of the conical part = √7225
Slant height of the conical part = 85 mm

Calculating the Height of the Cylindrical Part

To find the height of the cylindrical part, we need to subtract the height of the conical part from the total height of the toy.

Height of the cylindrical part = Total height - Height of the conical part
Height of the cylindrical part = 110 mm - 77 mm
Height of the cylindrical part = 33 mm

Calculating the Volume of the Conical Part

The volume of a cone can be calculated using the formula:

Volume of a cone = (1/3) * π * radius^2 * height

We can plug in the values to find the volume of the conical part.

Volume of the conical part = (1/3) * π * 36^2 * 77
Volume of the conical part = (1/3) * 3.14159 * 1296 * 77
Volume of the conical part = 310,111.19 mm^3

Calculating the Volume of the Cylindrical Part

The volume of a cylinder can be calculated using the formula:

Volume of a cylinder = π * radius^2 * height

We can plug in the values to find the volume of the cylindrical part.

Volume of the cylindrical part = π * 20^2 * 33
Volume of the cylindrical part = 3.14159 * 400 * 33
Volume of the cylindrical part = 131,704.19 mm^3

Calculating the Total Volume of the Toy

To find the total volume of the toy, we need to add the volumes of the conical and cylindrical parts.

Total volume of the toy = Volume of the conical part + Volume of the cylindrical part
Total volume of the toy = 310,111.19 mm^3 + 131,704.19 mm^3
Total volume of the toy = 441,815.38 mm^3

Conclusion

Q: What is the shape of the unique toy?

A: The unique toy consists of a cone mounted on a cylinder.

Q: What are the dimensions of the conical part of the toy?

A: The height of the conical part is 77 mm, and the diameter of its base is 72 mm.

Q: What are the dimensions of the cylindrical part of the toy?

A: The height of the cylindrical part is 33 mm, and the diameter of its base is 40 mm.

Q: How can we calculate the radius of the conical part?

A: To calculate the radius of the conical part, we need to divide its diameter by 2. So, the radius of the conical part is 36 mm.

Q: How can we calculate the radius of the cylindrical part?

A: To calculate the radius of the cylindrical part, we need to divide its diameter by 2. So, the radius of the cylindrical part is 20 mm.

Q: How can we calculate the slant height of the conical part?

A: We can use the Pythagorean theorem to calculate the slant height of the conical part. The slant height of the conical part is √(77^2 + 36^2) = 85 mm.

Q: How can we calculate the volume of the conical part?

A: We can use the formula: Volume of a cone = (1/3) * π * radius^2 * height. Plugging in the values, we get: Volume of the conical part = (1/3) * π * 36^2 * 77 = 310,111.19 mm^3.

Q: How can we calculate the volume of the cylindrical part?

A: We can use the formula: Volume of a cylinder = π * radius^2 * height. Plugging in the values, we get: Volume of the cylindrical part = π * 20^2 * 33 = 131,704.19 mm^3.

Q: How can we calculate the total volume of the toy?

A: To find the total volume of the toy, we need to add the volumes of the conical and cylindrical parts. So, the total volume of the toy is 310,111.19 mm^3 + 131,704.19 mm^3 = 441,815.38 mm^3.

Q: What is the significance of the unique toy's geometry?

A: The unique toy's geometry provides a practical application of mathematical concepts, such as the Pythagorean theorem and the formulas for the volume of a cone and a cylinder.

Q: Can we use the unique toy's geometry to solve real-world problems?

A: Yes, the unique toy's geometry can be used to solve real-world problems, such as calculating the volume of a container or the height of a building.

Q: How can we apply the concepts learned from the unique toy's geometry to other areas of mathematics?

A: The concepts learned from the unique toy's geometry can be applied to other areas of mathematics, such as trigonometry, calculus, and engineering.