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: Compare the cone directly to the cube's volume, using a ratio, instead of computing the cone's volume from scratch.
The largest cone that fits inside a cube of edge $l$ has its circular base inscribed in one face of the cube, so its radius is $r = \frac{l}{2}$, and its height equals the cube's edge, $h = l$, since the cone's apex just touches the opposite face.

Step 2: Write the ratio of the cone's volume to the cube's volume.
The cube's volume is simply $l^3$. Write the ratio:
\[ \frac{V_{\text{cone}}}{V_{\text{cube}}} = \frac{\frac{1}{3}\pi r^2 h}{l^3} \]

Step 3: Substitute r and h in terms of l, and simplify the ratio first before multiplying out.
\[ \frac{V_{\text{cone}}}{V_{\text{cube}}} = \frac{\frac{1}{3}\pi \left(\frac{l}{2}\right)^2 (l)}{l^3} = \frac{\frac{1}{3}\pi \cdot \frac{l^2}{4} \cdot l}{l^3} = \frac{\frac{\pi l^3}{12}}{l^3} \]
Cancel $l^3$ from numerator and denominator:
\[ \frac{V_{\text{cone}}}{V_{\text{cube}}} = \frac{\pi}{12} \]

Step 4: Multiply back by the cube's volume to get the cone's actual volume.
\[ V_{\text{cone}} = \frac{\pi}{12} \times l^3 = \frac{\pi l^3}{12} \]

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