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: Find the volume of the cylinder that would fit in the same cube.
The largest cylinder that fits inside a cube of edge $l$ has its circular base inscribed in a face of the cube, so its radius is $r=\dfrac{l}{2}$ and its height equals the edge, $h=l$. Its volume is:
\[ V_{\text{cyl}} = \pi r^{2}h = \pi\left(\frac{l}{2}\right)^{2}(l) = \frac{\pi l^{3}}{4} \]

Step 2: Use the known ratio between a cone and a cylinder of the same base and height.
A cone with the same base radius and height as a cylinder always has exactly one third of the cylinder's volume:
\[ V_{\text{cone}} = \frac{1}{3}V_{\text{cyl}} \]

Step 3: Substitute to get the cone's volume.
\[ V_{\text{cone}} = \frac{1}{3}\left(\frac{\pi l^{3}}{4}\right) = \frac{\pi l^{3}}{12} \]

Final Answer:
The volume of the largest cone that can be carved from the cube is $\dfrac{\pi l^{3}}{12}$, matching option (A).
\[ \boxed{V = \frac{\pi l^{3}}{12}} \]