Question

A cone of maximum size is carved out from a solid cube of edge length \(l\). The volume of the cone is :

• A. \(\frac{\pi l^3}{12}\)
• B. \(\frac{\pi l^3}{3}\)
• C. \(l^3\left(1 - \frac{\pi}{3}\right)\)
• D. \(\frac{\pi l^3}{8}\)

Hint:

A useful shortcut relation to remember:
The volume of a maximum size cylinder carved out from a cube of side \( l \) is \( \frac{\pi l^3}{4} \).
The volume of a maximum size cone is exactly \(\frac{1}{3}\) of this cylinder's volume:
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{\pi l^3}{4} = \frac{\pi l^3}{12} \]
Remembering these proportional relationships helps you solve direct multiple-choice questions quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Surface Areas and Volumes.
We are given a solid cube of edge length \( l \).
We need to find the volume of a cone of the maximum possible size that can be carved out from this cube.

Step 2: Key Formula or Approach:
- The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the circular base of the cone.
- \( h \) is the height of the cone.
- For a cone to have the maximum possible volume inside a cube of side \( l \):
1. Its circular base must be inscribed in one of the square faces of the cube.
Therefore, the diameter of the circular base of the cone must be equal to the side length of the cube.
\[ \text{Diameter} = 2r = l \implies r = \frac{l}{2} \]
2. Its height must be equal to the vertical height of the cube.
\[ h = l \]

Step 3: Detailed Explanation:
1. Determine the maximum dimensions of the cone that can fit inside the cube of side length \( l \):
- Radius of the cone, \( r = \frac{l}{2} \)
- Height of the cone, \( h = l \)
2. Use the volume formula for the cone:
\[ V = \frac{1}{3} \pi r^2 h \]
3. Substitute the values of \( r \) and \( h \) in terms of \( l \):
\[ V = \frac{1}{3} \pi \left(\frac{l}{2}\right)^2 (l) \]
4. Simplify the squared radius term:
\[ \left(\frac{l}{2}\right)^2 = \frac{l^2}{4} \]
5. Substitute this back into the volume equation:
\[ V = \frac{1}{3} \pi \left(\frac{l^2}{4}\right) (l) \]
6. Multiply the constants and variables:
\[ V = \frac{\pi l^3}{12} \]
7. Thus, the volume of the carved-out cone of maximum size is \(\frac{\pi l^3}{12}\).

Step 4: Final Answer:
The volume of the cone of maximum size is \(\frac{\pi l^3}{12}\), which matches option (A).