Finding the Volume of the Toy

Oct 13, 2025 by windailys.com 113 views

Introduction

In this problem, we are given a toy in the shape of a hemisphere on the base of a cone. The total height of the toy is 12 cm, and the common radius is 3 cm. Our goal is to find the volume of this unique toy. To solve this problem, we will use the concepts of volumes of cones and hemispheres.

Understanding the Shape

The toy is a combination of a cone and a hemisphere. The cone has a radius of 3 cm and a height of 9 cm (since the total height is 12 cm, and the hemisphere has a height of 3 cm). The hemisphere has a radius of 3 cm.

Volume of a Cone

The formula for the volume of a cone is given by:

V = (1/3)πr²h

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

Volume of a Hemisphere

The formula for the volume of a hemisphere is given by:

V = (2/3)πr³

where V is the volume, π is a constant approximately equal to 3.14, and r is the radius.

Finding the Volume of the Toy

To find the volume of the toy, we need to find the volume of the cone and the hemisphere separately and then add them together.

Volume of the Cone

Using the formula for the volume of a cone, we get:

V_cone = (1/3)π(3)²(9) = (1/3)π(9)(9) = 27π

Volume of the Hemisphere

Using the formula for the volume of a hemisphere, we get:

V_hemisphere = (2/3)π(3)³ = (2/3)π(27) = 18π

Total Volume of the Toy

The total volume of the toy is the sum of the volume of the cone and the hemisphere:

V_total = V_cone + V_hemisphere = 27π + 18π = 45π

Conclusion

In this problem, we found the volume of a toy in the shape of a hemisphere on the base of a cone. The total height of the toy is 12 cm, and the common radius is 3 cm. We used the formulas for the volumes of cones and hemispheres to find the volume of the toy. The total volume of the toy is 45π cubic centimeters.

Final Answer

The final answer is 45π cubic centimeters.

Additional Information

The volume of the toy can be approximated as 141.37 cubic centimeters using the value of π as 3.14.
The volume of the toy can be calculated using a calculator or a computer program for more accurate results.
The shape of the toy is a combination of a cone and a hemisphere, which makes it a unique and interesting shape.

References

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

Introduction

In our previous article, we discussed how to find the volume of a toy in the shape of a hemisphere on the base of a cone. In this article, we will answer some frequently asked questions related to this problem.

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

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

V = (1/3)πr²h

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

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

A: The formula for the volume of a hemisphere is given by:

V = (2/3)πr³

where V is the volume, π is a constant approximately equal to 3.14, and r is the radius.

Q: How do I find the volume of a cone with a radius of 4 cm and a height of 6 cm?

A: To find the volume of a cone with a radius of 4 cm and a height of 6 cm, you can use the formula for the volume of a cone:

V = (1/3)π(4)²(6) = (1/3)π(16)(6) = 32π

Q: How do I find the volume of a hemisphere with a radius of 5 cm?

A: To find the volume of a hemisphere with a radius of 5 cm, you can use the formula for the volume of a hemisphere:

V = (2/3)π(5)³ = (2/3)π(125) = 83.33π

Q: How do I find the volume of a sphere with a radius of 6 cm?

A: To find the volume of a sphere with a radius of 6 cm, you can use the formula for the volume of a sphere:

V = (4/3)πr³

where V is the volume, π is a constant approximately equal to 3.14, and r is the radius.

V = (4/3)π(6)³ = (4/3)π(216) = 288π

Q: What is the difference between the volume of a cone and the volume of a hemisphere?

A: The volume of a cone is given by:

V = (1/3)πr²h

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

The volume of a hemisphere is given by:

V = (2/3)πr³

where V is the volume, π is a constant approximately equal to 3.14, and r is the radius.

The main difference between the two formulas is the power of the radius. The volume of a cone is proportional to the square of the radius, while the volume of a hemisphere is proportional to the cube of the radius.

Q: Can I use a calculator or a computer program to find the volume of a cone or a hemisphere?

A: Yes, you can use a calculator or a computer program to find the volume of a cone or a hemisphere. Many calculators and computer programs have built-in functions for calculating the volume of cones and hemispheres.

Conclusion

In this article, we answered some frequently asked questions related to finding the volume of a cone and a hemisphere. We also provided examples of how to use the formulas for the volume of a cone and a hemisphere. We hope this article has been helpful in understanding the concepts of volumes of cones and hemispheres.

Final Answer

The final answer is that the volume of a cone and a hemisphere can be found using the formulas:

V = (1/3)πr²h

V = (2/3)πr³

respectively.

Additional Information

The volume of a cone and a hemisphere can be approximated using a calculator or a computer program.
The formulas for the volume of a cone and a hemisphere can be used to find the volume of other shapes, such as spheres and cylinders.
The concepts of volumes of cones and hemispheres are important in many fields, including engineering, physics, and mathematics.

References

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