Calculating the Volume of the Largest Cone from a Cube

The problem asks us to find the volume of the largest possible right circular cone that can be formed by cutting it from a cube with side length 8 cm. We are given the value of pi as $\frac{22}{7}$.

Understanding the Geometry

To cut the largest possible right circular cone from a cube:

• The diameter of the cone's base must be equal to the edge length of the cube.
• The height of the cone must also be equal to the edge length of the cube.

Identifying Dimensions

Given:

• Edge length of the cube, denoted as '$a$' = 8 cm.

For the largest inscribed cone:

• Diameter of the base ($d$) = $a$ = 8 cm.
• Radius of the base ($r$) = $\frac{d}{2} = \frac{8 \text{ cm}}{2}$ = 4 cm.
• Height of the cone ($h$) = $a$ = 8 cm.

Applying the Volume Formula

The formula for the volume ($V$) of a right circular cone is:

$$V = \frac{1}{3} \pi r^2 h$$

Step-by-Step Calculation

1. Substitute the known values into the formula: $$V = \frac{1}{3} \times \frac{22}{7} \times (4 \text{ cm})^2 \times (8 \text{ cm})$$
2. Calculate the square of the radius: $$(4 \text{ cm})^2 = 16 \text{ cm}^2$$
3. Multiply the terms: $$V = \frac{1}{3} \times \frac{22}{7} \times 16 \text{ cm}^2 \times 8 \text{ cm}$$ $$V = \frac{1}{3} \times \frac{22}{7} \times 128 \text{ cm}^3$$
4. Perform the multiplication: $$V = \frac{22 \times 128}{3 \times 7} \text{ cm}^3$$ $$V = \frac{2816}{21} \text{ cm}^3$$
5. Convert the improper fraction to a mixed number: Divide 2816 by 21: $$2816 \div 21 = 134 \text{ with a remainder of } 2$$ So, the volume is $134\frac{2}{21}$ cm³.

Final Volume

The volume of the largest right circular cone that can be cut from the cube is $\frac{2816}{21}$ cm³, which is equal to $134\frac{2}{21}$ cm³.