Relationship Between the Cylinder and the Sphere

Oct 13, 2025 by miodeya.com 89 views

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Introduction

In geometry, understanding the relationship between different shapes is crucial for solving various mathematical problems. In this article, we will explore the relationship between a sphere and a cylinder when they have the same radius and height. We will use the given volume of the cylinder to find the volume of the sphere.

Volume of a Cylinder

The volume of a cylinder is given by the formula:

V_cylinder = πr^2h

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

Volume of a Sphere

The volume of a sphere is given by the formula:

V_sphere = (4/3)πr^3

where r is the radius of the sphere.

Given Information

We are given that the volume of the cylinder is 50 ft^3. Since the cylinder and the sphere have the same radius and height, we can use this information to find the volume of the sphere.

Relationship Between the Cylinder and the Sphere

Since the cylinder and the sphere have the same radius and height, we can set up a proportion to relate their volumes. Let's denote the radius and height of both the cylinder and the sphere as r and h, respectively.

V_cylinder = V_sphere

Substituting the formulas for the volumes of the cylinder and the sphere, we get:

πr^2h = (4/3)πr^3

Simplifying the Equation

We can simplify the equation by canceling out the π term and dividing both sides by r^2:

h = (4/3)r

Finding the Volume of the Sphere

Now that we have the relationship between the height and the radius of the sphere, we can substitute this expression into the formula for the volume of the sphere:

V_sphere = (4/3)πr^3

Substituting h = (4/3)r into the formula, we get:

V_sphere = (4/3)πr^3 = (4/3)π((3/4)h)^3 = (4/3)π(27/64)h^3 = (27/64)πh^3

Substituting the Given Volume of the Cylinder

We are given that the volume of the cylinder is 50 ft^3. Since the cylinder and the sphere have the same radius and height, we can substitute this value into the formula for the volume of the sphere:

V_sphere = (27/64)πh^3 = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3))

Simplifying the Expression

We can simplify the expression by canceling out the π term and dividing both sides by r^2:

V_sphere = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3))

Evaluating the Expression

Evaluating the expression, we get:

V_sphere = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3))

Conclusion

In conclusion, we have found that the volume of the sphere is (27/64)π((50/πr²⁾(2/3)). However, we can simplify this expression further by canceling out the π term and dividing both sides by r^2:

V_sphere = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3)) = (27/64)π((50/πr²⁾(2/3))

Final Answer

The final answer is $\boxed{\frac{100}{3} \, \text{ft}^3}$ .

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Q&A: A Sphere and a Cylinder with the Same Radius and Height

Q: What is the relationship between the volume of a cylinder and a sphere with the same radius and height?

A: The volume of a cylinder is given by the formula V_cylinder = πr^2h, while the volume of a sphere is given by the formula V_sphere = (4/3)πr^3. Since the cylinder and the sphere have the same radius and height, we can set up a proportion to relate their volumes.

Q: How do we find the volume of the sphere when the volume of the cylinder is given?

A: We can substitute the expression for the height of the sphere (h = (4/3)r) into the formula for the volume of the sphere (V_sphere = (4/3)πr^3). This gives us V_sphere = (27/64)πh^3.

Q: What is the final answer for the volume of the sphere?

A: The final answer is $\boxed{\frac{100}{3} \, \text{ft}^3}$ .

Q: Why is the volume of the sphere larger than the volume of the cylinder?

A: The volume of the sphere is larger than the volume of the cylinder because the sphere has a larger volume-to-surface-area ratio. This means that the sphere has a larger volume than the cylinder for the same surface area.

Q: Can we use this relationship to find the volume of a sphere when the volume of a cylinder is given?

A: Yes, we can use this relationship to find the volume of a sphere when the volume of a cylinder is given. We simply need to substitute the expression for the height of the sphere (h = (4/3)r) into the formula for the volume of the sphere (V_sphere = (4/3)πr^3).

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

A: This relationship has many real-world applications, such as in engineering and architecture. For example, when designing a building or a bridge, engineers need to consider the volume of the structure and the surface area of the structure. This relationship can help them make informed decisions about the design of the structure.

Q: Can we use this relationship to find the volume of a cylinder when the volume of a sphere is given?

A: Yes, we can use this relationship to find the volume of a cylinder when the volume of a sphere is given. We simply need to substitute the expression for the radius of the cylinder (r = (3/4)h) into the formula for the volume of the cylinder (V_cylinder = πr^2h).

Q: What are some limitations of this relationship?

A: One limitation of this relationship is that it assumes that the cylinder and the sphere have the same radius and height. In reality, the cylinder and the sphere may have different dimensions, which can affect the relationship between their volumes.

Q: Can we use this relationship to find the volume of a sphere when the volume of a cylinder is given, but the radius and height of the cylinder are not given?

A: No, we cannot use this relationship to find the volume of a sphere when the volume of a cylinder is given, but the radius and height of the cylinder are not given. We need to know the radius and height of the cylinder in order to use this relationship.

Q: What are some other formulas that we can use to find the volume of a sphere?

A: There are many other formulas that we can use to find the volume of a sphere, such as the formula V_sphere = (4/3)πr^3. We can also use the formula V_sphere = (4/3)π(3/4)h^3, which is derived from the formula for the volume of a cylinder.

Q: Can we use this relationship to find the volume of a cylinder when the volume of a sphere is given, but the radius and height of the sphere are not given?

A: No, we cannot use this relationship to find the volume of a cylinder when the volume of a sphere is given, but the radius and height of the sphere are not given. We need to know the radius and height of the sphere in order to use this relationship.

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

A: This relationship has many other real-world applications, such as in physics and engineering. For example, when designing a machine or a system, engineers need to consider the volume of the system and the surface area of the system. This relationship can help them make informed decisions about the design of the system.

Q: Can we use this relationship to find the volume of a sphere when the volume of a cylinder is given, but the radius and height of the cylinder are not given?

Q: What are some other formulas that we can use to find the volume of a cylinder?

A: There are many other formulas that we can use to find the volume of a cylinder, such as the formula V_cylinder = πr^2h. We can also use the formula V_cylinder = π(3/4)h^3, which is derived from the formula for the volume of a sphere.

Q: Can we use this relationship to find the volume of a cylinder when the volume of a sphere is given, but the radius and height of the sphere are not given?