Question

If a cone of greatest possible volume is hollowed out from a solid wooden cylinder, then the ratio of the volume of remaining wood to the volume of cone hollowed out is

• A. 1 : 2
• B. 2 : 1
• C. 1 : 1
• D. 3 : 1

Hint:

Remember: volume of cone = $\dfrac{1}{3}$ of volume of cylinder.

Answer 1

The Correct Option is B

Problem:
A cone of maximum volume is removed from a solid wooden cylinder. Determine the ratio of the remaining wood's volume to the cone's volume.

Solution:
Let \(r\) be the radius and \(h\) the height of both the cylinder and cone.

Volume of cylinder: \(\pi r^2 h\)
Volume of cone: \(\dfrac{1}{3} \pi r^2 h\)

Remaining volume = Volume of cylinder - Volume of cone = \(\pi r^2 h - \dfrac{1}{3} \pi r^2 h = \dfrac{2}{3} \pi r^2 h\)

\[\text{Ratio} = \frac{\text{Remaining volume}}{\text{Cone volume}} = \frac{\dfrac{2}{3} \pi r^2 h}{\dfrac{1}{3} \pi r^2 h} = \frac{2}{1}\]

Answer:
The ratio is 2 : 1.