The Given Volume of the Cylinder

Oct 13, 2025 by miodeya.com 69 views

When dealing with geometric shapes, it's essential to understand the relationships between their volumes. In this article, we'll explore the relationship between the volume of a sphere and a cylinder, given that they have the same radius and height.

The Volume of a Cylinder

The volume of a cylinder is given by the formula:

V = πr²h

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

The Given Volume of the Cylinder

We're given that the volume of the cylinder is $27 \pi t^3$ . This can be written as:

V = 27πt³

The Relationship Between the Volume of a Sphere and a Cylinder

The volume of a sphere is given by the formula:

V = (4/3)πr³

where V is the volume, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

Since the sphere and the cylinder have the same radius and height, we can set up a relationship between their volumes. Let's start by finding the radius of the cylinder.

Finding the Radius of the Cylinder

We know that the volume of the cylinder is $27 \pi t^3$ . We can set up an equation using the formula for the volume of a cylinder:

V = πr²h

Substituting the given volume, we get:

27πt³ = πr²h

Since the height of the cylinder is equal to the height of the sphere, we can substitute h = t. The equation becomes:

27πt³ = πr²t

Dividing both sides by πt, we get:

27t² = r²

Taking the square root of both sides, we get:

r = √(27t²)

Simplifying, we get:

r = 3√(3t²)

Finding the Volume of the Sphere

Now that we have the radius of the cylinder, we can find the volume of the sphere. We know that the volume of a sphere is given by the formula:

V = (4/3)πr³

Substituting the value of r, we get:

V = (4/3)π(3√(3t²))³

Expanding the cube, we get:

V = (4/3)π(27(3t²)²)

Simplifying, we get:

V = (4/3)π(27(9t⁴))

V = (4/3)π(243t⁴)

V = 81πt⁴

Comparing the Options

Now that we have the volume of the sphere, let's compare it with the given options.

Option A: $V = \frac{2}{3}(27 \pi t^3)$

This option is incorrect because the volume of the sphere is not equal to $\frac{2}{3}$ times the volume of the cylinder.

Option B: $V = -407$

This option is also incorrect because the volume of the sphere cannot be negative.

Conclusion

In conclusion, the volume of the sphere is given by the equation:

V = 81πt⁴

This equation is derived from the relationship between the volume of a sphere and a cylinder, given that they have the same radius and height.

Discussion

What do you think about the relationship between the volume of a sphere and a cylinder? Have you encountered any similar problems in your math studies? Share your thoughts and experiences in the comments below!

References

[1] "Mathematics for Dummies" by Mary Jane Sterling
[2] "Geometry: A Comprehensive Introduction" by Dan Pedoe

[1] "The Volume of a Sphere: A Comprehensive Guide"
[2] "The Relationship Between the Volume of a Cylinder and a Cone"
[3] "Mathematical Constants: A Brief Introduction to Pi and Euler's Number"
A Sphere and a Cylinder: Q&A ================================

In our previous article, we explored the relationship between the volume of a sphere and a cylinder, given that they have the same radius and height. We derived the equation for the volume of the sphere and compared it with the given options. In this article, we'll answer some frequently asked questions related to this topic.

Q: What is the relationship between the volume of a sphere and a cylinder?

A: The volume of a sphere is given by the formula:

V = (4/3)πr³

where V is the volume, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

The volume of a cylinder is given by the formula:

V = πr²h

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

Q: How do we find the radius of the cylinder?

A: To find the radius of the cylinder, we can use the given volume of the cylinder:

V = 27πt³

We can set up an equation using the formula for the volume of a cylinder:

27πt³ = πr²h

Substituting h = t, we get:

27πt³ = πr²t

Dividing both sides by πt, we get:

27t² = r²

Taking the square root of both sides, we get:

r = √(27t²)

Simplifying, we get:

r = 3√(3t²)

Q: How do we find the volume of the sphere?

A: Now that we have the radius of the cylinder, we can find the volume of the sphere. We know that the volume of a sphere is given by the formula:

V = (4/3)πr³

Substituting the value of r, we get:

V = (4/3)π(3√(3t²))³

Expanding the cube, we get:

V = (4/3)π(27(3t²)²)

Simplifying, we get:

V = (4/3)π(27(9t⁴))

V = (4/3)π(243t⁴)

V = 81πt⁴

Q: What is the correct equation for the volume of the sphere?

A: The correct equation for the volume of the sphere is:

V = 81πt⁴

This equation is derived from the relationship between the volume of a sphere and a cylinder, given that they have the same radius and height.

Q: What are some common mistakes to avoid when solving this problem?

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

Not using the correct formula for the volume of a sphere
Not substituting the correct value of r into the formula for the volume of the sphere
Not simplifying the equation correctly
Not checking the units of the answer

Q: How can I apply this concept to real-world problems?

A: This concept can be applied to real-world problems in various fields, such as engineering, physics, and architecture. For example, you may need to calculate the volume of a sphere or a cylinder in a design or a construction project.

Conclusion

In conclusion, the relationship between the volume of a sphere and a cylinder is an important concept in mathematics and physics. By understanding this relationship, we can derive the equation for the volume of the sphere and apply it to real-world problems.

Discussion

References

[1] "Mathematics for Dummies" by Mary Jane Sterling
[2] "Geometry: A Comprehensive Introduction" by Dan Pedoe

[1] "The Volume of a Sphere: A Comprehensive Guide"
[2] "The Relationship Between the Volume of a Cylinder and a Cone"
[3] "Mathematical Constants: A Brief Introduction to Pi and Euler's Number"