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:

Visualize the cone's base touching all four sides of the bottom face of the cube.
This establishes that the diameter is exactly \(l\).
The apex of the cone touches the center of the opposite face, making the height exactly \(l\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A cone of the largest possible size is carved out from a solid cube whose edge length is \(l\).
We need to express the volume of this maximum-sized cone in terms of \(l\).

Step 2: Key Formula or Approach:
For a cone to have the maximum size inside a cube of side \(l\):
- Its base must fit perfectly on one of the square faces of the cube. Thus, the diameter of the circular base of the cone is equal to the edge length of the cube: \(2r = l \implies r = \frac{l}{2}\).
- Its height must be equal to the height of the cube: \(h = l\).
The volume \(V\) of a cone is given by:
\[ V = \frac{1}{3}\pi r^2 h \]

Step 3: Detailed Explanation:
1. Identify the dimensions of the largest cone that can be carved out from a cube of side \(l\):
- Radius of the cone's base, \(r = \frac{l}{2}\)
- Height of the cone, \(h = l\)
2. Substitute these values into the volume formula of the cone:
\[ V = \frac{1}{3} \pi \left(\frac{l}{2}\right)^2 (l) \] 3. Simplify the squared term:
\[ \left(\frac{l}{2}\right)^2 = \frac{l^2}{4} \] 4. Substitute this back into the volume expression:
\[ V = \frac{1}{3} \pi \left(\frac{l^2}{4}\right) (l) \] 5. Multiply the terms:
\[ V = \frac{\pi l^3}{12} \] 6. Therefore, the volume of the maximum-sized carved-out cone is \(\frac{\pi l^3}{12}\).

Step 4: Final Answer:
The correct option is (A).