Question

A solid wooden cylinder is of height h cm and radius r cm. A conical cavity of the same height and the same radius is drilled out of the solid cylinder.

Statement (1): The volume of the remaining wood = volume of solid cylinder − volume of the cone drilled.

Statement (2): Volume of the remaining wood \[= \pi r^2 h - \frac{1}{3}, \pi r^2 h = \frac{2}{3}, \pi r^2 h.\]

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

The volume of the remaining wood is the volume of the original solid cylinder minus the volume of the conical cavity drilled out.

Let radius and height of cylinder and conical cavity (both are equal) be \[r\] and \[h\] units.

Volume of the cylinder : \[\pi r^2 h\]

Volume of the cone : \[\frac{1}{3}\, \pi r^2 h\]

The volume of the remaining wood = Volume of the solid cylinder − Volume of the cone drilled.

\[{} = \pi r^2 h - \frac{1}{3}\, \pi r^2 h\]

\[{} = \left(1 - \frac{1}{3}\right) \pi r^2 h\]

\[{} = \left(\frac{3 - 1}{3}\right) \pi r^2 h\]

\[{} = \frac{2}{3}\, \pi r^2 h.\]

∴ Both the statements are true.