Finding the Volume of the Largest Sphere Carved from a Cube

Let's break down this geometry problem step by step. We are given a cube with a side length of 6 cm, and we need to find the volume of the largest possible sphere that can be carved out of it.

Understanding the Relationship Between the Cube and the Sphere

When the largest possible sphere is carved from a cube, the key relationship is that the diameter of the sphere must be equal to the side length of the cube. Any larger sphere would not fit entirely within the cube's boundaries.

• Side length of the cube = 6 cm
• Diameter of the largest sphere = Side length of the cube = 6 cm
• Radius of the sphere = Diameter / 2 = 6 cm / 2 = 3 cm

So, the radius of the sphere is 3 cm.

Calculating the Sphere's Volume

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

\(V = \frac{4}{3}\pi r^3\)

where \(V\) is the volume and \(r\) is the radius. We can use the approximation \(\pi \approx \frac{22}{7}\) as the options are given in fractions involving 7.

Substitute the radius \(r = 3\) cm into the formula:

\(V = \frac{4}{3} \times \frac{22}{7} \times (3 \text{ cm})^3\)

Calculate the cube of the radius:

\((3 \text{ cm})^3 = 3 \times 3 \times 3 \text{ cm}^3 = 27 \text{ cm}^3\)

Now, substitute this back into the volume formula:

\(V = \frac{4}{3} \times \frac{22}{7} \times 27 \text{ cm}^3\)

We can simplify the calculation by cancelling out the 3 in the denominator with the 27:

\(V = 4 \times \frac{22}{7} \times \frac{27}{3} \text{ cm}^3\)

\(V = 4 \times \frac{22}{7} \times 9 \text{ cm}^3\)

Now, multiply the numbers together:

\(V = (4 \times 22 \times 9) \div 7 \text{ cm}^3\)

\(V = (88 \times 9) \div 7 \text{ cm}^3\)

\(V = 792 \div 7 \text{ cm}^3\)

So, the volume of the largest sphere that can be carved from the cube is \(\frac{792}{7}\) cm³.

Comparing with Options

The calculated volume is \(\frac{792}{7}\) cm³, which matches one of the given options.

Volume Calculation Steps Summary

1. Identify the cube's side length: 6 cm.
2. Determine the sphere's diameter (equal to the cube's side): 6 cm.
3. Calculate the sphere's radius (half the diameter): 3 cm.
4. Use the sphere volume formula: \(V = \frac{4}{3}\pi r^3\).
5. Substitute \(r=3\) and \(\pi \approx \frac{22}{7}\) into the formula.
6. Perform the calculation to find the volume.

Revision Table: Cube and Sphere Properties

Additional Information on Carving Shapes

When one shape is carved out of another, the dimensions of the carved shape are limited by the dimensions of the original shape. For carving the largest possible shape, the critical dimensions must match at their limits.

• Largest cylinder from a cube: The cylinder's diameter and height would both be equal to the cube's side length.
• Largest cube from a sphere: The vertices of the cube would lie on the sphere's surface. The diagonal of the cube would be equal to the diameter of the sphere.
• Largest sphere from a cylinder: If the cylinder's height is greater than or equal to its diameter, the sphere's diameter will equal the cylinder's diameter. If the cylinder's height is less than its diameter, the sphere's diameter will equal the cylinder's height.

These relationships are crucial for solving problems involving volumes and dimensions of shapes carved from other shapes.