Mensuration (Pyramid, Cones and Spheres) - Volume and Surface Area of Composite Solids

A solid consists of a cuboid and a rectangular-based pyramid joined along the cuboid’s entire top face (Diagram not drawn to scale). The cuboid measures \(12\text{ cm}\) by \(8\text{ cm}\) by \(7\text{ cm}\). The total vertical height of the solid is \(16\text{ cm}\).

Kai calculates the volume of the solid as
\[12\times8\times7+\frac13\times12\times8\times16.\]

1. Explain Kai’s mistake.

   [1]

2. Find the correct volume of the solid.

   [3]

3. The pyramid’s vertex is moved sideways while its base and perpendicular height remain unchanged. Does the volume of the solid change? Explain your answer.

   [1]