A ball is to be made with inner radius of 2 units and outside radius of 3 units. How much material is required to make the ball?
Calculating Material Needed for a Spherical Ball
This question asks us to determine the amount of material required to create a ball with a specific inner and outer radius. This shape is known as a spherical shell. To find the amount of material, we need to calculate the volume of this spherical shell.
A spherical shell is essentially a larger sphere with a smaller sphere removed from its center. Therefore, the volume of the material is the difference between the volume of the outer sphere and the volume of the inner sphere.
Understanding Sphere Volume
The formula for the volume of a sphere is:
\(V = \frac{4}{3}\pi r^3\)
where:
- \(V\) is the volume of the sphere
- \(\pi\) is a mathematical constant (approximately 3.14159)
- \(r\) is the radius of the sphere
Calculating the Volume of the Outer Sphere
The problem states the outside radius of the ball is 3 units. Let's call this \(R\).
\(R = 3\) units
The volume of the outer sphere (\(V_{outer}\)) is:
\(V_{outer} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (3^3)\)
\(V_{outer} = \frac{4}{3}\pi (27)\)
Calculating the Volume of the Inner Sphere
The problem states the inner radius of the ball is 2 units. Let's call this \(r\).
\(r = 2\) units
The volume of the inner sphere (\(V_{inner}\)) is:
\(V_{inner} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (2^3)\)
\(V_{inner} = \frac{4}{3}\pi (8)\)
Determining the Amount of Material
The amount of material required is the volume of the spherical shell, which is the volume of the outer sphere minus the volume of the inner sphere.
Amount of Material = \(V_{outer} - V_{inner}\)
Amount of Material = \(\frac{4}{3}\pi (27) - \frac{4}{3}\pi (8)\)
We can factor out \(\frac{4}{3}\pi\) from both terms:
Amount of Material = \(\frac{4}{3}\pi (27 - 8)\)
Amount of Material = \(\frac{4}{3}\pi (19)\)
Amount of Material = \(\frac{76}{3}\pi\)
So, the amount of material required to make the ball is \(\frac{76}{3}\pi\) cubic units.
Summary of Calculation Steps
| Step | Description | Formula/Calculation |
|---|---|---|
| 1 | Identify radii | Outer Radius (\(R\)) = 3, Inner Radius (\(r\)) = 2 |
| 2 | Calculate outer sphere volume | \(V_{outer} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (3^3) = \frac{4}{3}\pi (27)\) |
| 3 | Calculate inner sphere volume | \(V_{inner} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (2^3) = \frac{4}{3}\pi (8)\) |
| 4 | Subtract to find material volume | \(V_{material} = V_{outer} - V_{inner} = \frac{4}{3}\pi (27) - \frac{4}{3}\pi (8) = \frac{4}{3}\pi (19) = \frac{76}{3}\pi\) |
The calculated amount of material is \(\frac{76}{3}\pi\), which corresponds to one of the given options.
Revision Table: Key Concepts for Sphere Volume
| Concept | Description | Formula |
|---|---|---|
| Sphere | A perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. | N/A |
| Radius (\(r\)) | The distance from the center of a sphere to any point on its surface. | N/A |
| Volume of a Sphere | The amount of 3D space a sphere occupies. | \(V = \frac{4}{3}\pi r^3\) |
| Spherical Shell | The region between two concentric spheres of different radii. | \(V_{shell} = V_{outer} - V_{inner} = \frac{4}{3}\pi (R^3 - r^3)\) |
Additional Information: Applications of Sphere Volume Calculations
Calculating the volume of spheres and spherical shells has many practical applications in various fields:
- Engineering: Determining the volume of tanks, pipes, or ball bearings.
- Physics: Calculating the volume of celestial bodies or understanding fluid dynamics within spherical containers.
- Manufacturing: Estimating the amount of raw material needed for producing spherical objects like ball bearings, cannonballs, or even some types of sports equipment.
- Chemistry: Understanding the volume occupied by spherical molecules or particles.
- Medicine: Estimating the volume of organs or tumors that can be approximated as spheres.
Understanding how to calculate the volume of a spherical shell is useful when dealing with hollow spherical objects, where the thickness of the wall is important for determining weight or material cost.
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