Understanding the Problem: Volume of Remaining Solid

The problem asks us to find the volume of the solid that remains after a conical cavity is removed from a solid cube. We are given the dimensions of the cube and the conical cavity.

• Solid Cube: Side length = 7 cm.
• Conical Cavity: Height = 7 cm, Radius = 3 cm.

To find the volume of the remaining solid, we need to calculate the volume of the original cube and subtract the volume of the conical cavity.

Volume of Remaining Solid = Volume of Cube - Volume of Conical Cavity

Calculating Volume of the Cube

The formula for the volume of a cube with side length \(s\) is given by:

\( V_{cube} = s^3 \)

Given the side length of the cube is 7 cm:

\( V_{cube} = (7 \text{ cm})^3 \)

\( V_{cube} = 7 \times 7 \times 7 \text{ cm}^3 \)

\( V_{cube} = 343 \text{ cm}^3 \)

So, the volume of the solid cube is 343 cm\(^3\).

Calculating Volume of the Conical Cavity

The formula for the volume of a cone with radius \(r\) and height \(h\) is given by:

\( V_{cone} = \frac{1}{3}\pi r^2 h \)

Given the radius of the conical cavity is 3 cm and the height is 7 cm. We can use the approximation \(\pi \approx \frac{22}{7}\).

\( V_{cone} = \frac{1}{3} \times \frac{22}{7} \times (3 \text{ cm})^2 \times (7 \text{ cm}) \)

\( V_{cone} = \frac{1}{3} \times \frac{22}{7} \times 9 \text{ cm}^2 \times 7 \text{ cm} \)

\( V_{cone} = \frac{1}{3} \times 22 \times 9 \times \frac{7}{7} \text{ cm}^3 \)

\( V_{cone} = \frac{1}{3} \times 22 \times 9 \text{ cm}^3 \)

\( V_{cone} = 22 \times \frac{9}{3} \text{ cm}^3 \)

\( V_{cone} = 22 \times 3 \text{ cm}^3 \)

\( V_{cone} = 66 \text{ cm}^3 \)

The volume of the conical cavity is 66 cm\(^3\).

Finding Volume of Remaining Solid

Now, we subtract the volume of the conical cavity from the volume of the cube to find the volume of the remaining solid.

\( V_{remaining} = V_{cube} - V_{cone} \)

\( V_{remaining} = 343 \text{ cm}^3 - 66 \text{ cm}^3 \)

\( V_{remaining} = 277 \text{ cm}^3 \)

The volume of the remaining solid is 277 cm\(^3\).

Revision Table: Cube and Cone Volumes

Additional Information: Volumes of Solids and Composite Shapes

Understanding how to calculate the volumes of basic 3D shapes like cubes, cones, cylinders, spheres, etc., is fundamental in Mensuration. Problems often involve finding the volume of composite solids, which are formed by combining two or more basic shapes or by removing one shape from another.

• When shapes are combined, their volumes are added.
• When a shape is hollowed out or removed, its volume is subtracted from the volume of the original shape.

Common formulas to remember include:

• Volume of Cylinder: \(V = \pi r^2 h\)
• Volume of Sphere: \(V = \frac{4}{3}\pi r^3\)
• Volume of Hemisphere: \(V = \frac{2}{3}\pi r^3\)

Always ensure the units of measurement are consistent throughout the calculation. In this problem, all dimensions were in centimeters, resulting in the volume in cubic centimeters (cm\(^3\)).