A cone of maximum size is carved out from a solid cube of edge length \(l\). The volume of the cone is :
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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.
Step 1: Understand the Question:
This question is from Surface Areas and Volumes.
We start with a solid cube of edge length $l$ and cut out the biggest cone that can fit inside it.
We need the volume of that cone.
Step 2: Find the base radius and height of the cone.
The biggest circle that fits on a square face of side $l$ has diameter equal to $l$ (the circle just touches all four sides of the square).
So the base radius of the cone is
\[ r = \frac{l}{2} \]
The cone can rise the full height of the cube without poking out, so the height of the cone is
\[ h = l \]
Step 3: Find the base area of the cone.
The base of the cone is a circle of radius $r = \frac{l}{2}$, so its area is
\[ A = \pi r^2 = \pi \left(\frac{l}{2}\right)^2 = \frac{\pi l^2}{4} \]
We work this out as its own step because the volume rule for a cone is built directly from the base area, not from $r$ and $h$ separately.
Step 4: Apply the cone volume rule using this base area.
For any cone, the volume is one third of the base area times the height:
\[ V = \frac{1}{3} \times A \times h \]
Put in $A = \frac{\pi l^2}{4}$ and $h = l$:
\[ V = \frac{1}{3} \times \frac{\pi l^2}{4} \times l \]
Multiply the base area by the height first:
\[ \frac{\pi l^2}{4} \times l = \frac{\pi l^3}{4} \]
Now take one third of that:
\[ V = \frac{1}{3} \times \frac{\pi l^3}{4} = \frac{\pi l^3}{12} \]
Final Answer:
The volume of the cone is $\frac{\pi l^3}{12}$, which matches option (A).
\[ \boxed{V = \frac{\pi l^3}{12}} \]
