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

Given:
A cone of greatest possible volume is hollowed out from a solid wooden cylinder.
We are to find the ratio of the volume of the remaining wood to the volume of the cone hollowed out.

Step 1: Use formulas for volumes
Let the radius of both the cylinder and cone be \(r\), and the height be \(h\) (since cone is carved out from the cylinder, their dimensions will be the same for maximum volume).

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

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

Step 3: Find the required ratio
\[ \text{Ratio} = \frac{\text{Remaining volume}}{\text{Cone volume}} = \frac{\dfrac{2}{3} \pi r^2 h}{\dfrac{1}{3} \pi r^2 h} = \frac{2}{1} \]
Final Answer:
The ratio is 2 : 1.