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:

For any cone of maximum size inscribed in a cube of side $l$:
The ratio of the volume of the cone to the volume of the cube is:
\[ \frac{V_{\text{cone}}}{V_{\text{cube}}} = \frac{\frac{\pi l^3}{12}}{l^3} = \frac{\pi}{12} \approx 26.18\% \] Remembering this ratio $\frac{\pi}{12}$ helps you immediately write down the answer for any variable representing the edge length of the cube.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question is from the topic "Surface Areas and Volumes".
We are carving out the largest possible cone from a solid cube of edge length $l$.
We need to determine the mathematical expression for the volume of this maximum-sized cone in terms of the edge length $l$.

Step 2: Key Formula or Approach:
To maximize the size of a cone carved from a cube:

• The base of the cone must be inscribed within one of the square faces of the cube.
Therefore, the diameter ($d$) of the circular base of the cone is equal to the edge length of the cube:
\[ d = l \implies \text{Radius } (r) = \frac{l}{2} \]

• The height ($h$) of the cone must extend fully from the base face of the cube to the opposite face.
Therefore, the height of the cone is equal to the edge length of the cube:
\[ h = l \]

• The standard formula for the volume ($V$) of a cone is:
\[ V = \frac{1}{3}\pi r^2 h \]

Step 3: Detailed Explanation:

• Let the side length of the solid cube be $l$.

• The circular base of the cone is inscribed inside a square face of side $l$.
The maximum diameter possible for this circle is $l$.
Thus, the radius $r$ of the cone is:
\[ r = \frac{l}{2} \]

• The maximum height $h$ that the cone can have within the boundaries of the cube is equal to the distance between opposite faces, which is the cube's edge length:
\[ h = l \]

• Substitute these values of $r$ and $h$ into the formula for the volume of a cone:
\[ V = \frac{1}{3}\pi \left(\frac{l}{2}\right)^2 (l) \]

• Simplify the squared term:
\[ \left(\frac{l}{2}\right)^2 = \frac{l^2}{4} \]

• Now multiply the terms together:
\[ V = \frac{1}{3} \cdot \pi \cdot \frac{l^2}{4} \cdot l \] \[ V = \frac{\pi l^3}{12} \]

Step 4: Final Answer:
The volume of the maximum cone that can be carved out is $\frac{\pi l^3}{12}$, which corresponds to Option (A).