Find the volume of that largest cone that can be cut from a cube of edge 14 cm.

Answer

To find the volume of the largest cone that can be cut from a cube with an edge length of 14 cm, we can follow these steps:

### Step 1: Determine the radius of the base of the cone
The largest cone that can be inscribed in the cube will have its base on one face of the cube. The diameter of the base of the cone will be equal to the edge of the cube.

- **Radius (r)** = Diameter / 2 = Edge of the cube / 2
- Given the edge of the cube = 14 cm, we have:
\[ r = \frac{14}{2} = 7 \text{ cm} \]

### Step 2: Determine the height of the cone
The height of the cone will be equal to the edge of the cube since the cone extends from the base to the top of the cube.

- **Height (h)** = Edge of the cube = 14 cm

### Step 3: Use the formula for the volume of a cone
The formula for the volume (V) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( \pi \approx \frac{22}{7} \)
- \( r = 7 \text{ cm} \)
- \( h = 14 \text{ cm} \)

### Step 4: Substitute the values into the formula
Substituting the values of \( r \) and \( h \) into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times (7^2) \times 14 \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Now substituting this back into the equation:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \times 14 \]

### Step 5: Simplify the expression
First, simplify \( \frac{49 \times 14}{7} \):
\[ \frac{49 \times 14}{7} = 49 \times 2 = 98 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times 22 \times 98 \]

### Step 6: Calculate the final volume
Now calculate \( 22 \times 98 \):
\[ 22 \times 98 = 2156 \]
Now divide by 3:
\[ V = \frac{2156}{3} \approx 718.67 \text{ cm}^3 \]

### Final Answer
The volume of the largest cone that can be cut from a cube of edge 14 cm is approximately:
\[ \boxed{718.67 \text{ cm}^3} \]