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The problem asks for the volume and total surface area of the largest cube of wood that can be cut from a log with a circular cross-section of radius 12.7 cm.

### Step-by-Step Solution:

1. **Understanding the Geometry:**
   - The cube is inscribed inside the circular cross-section of the log. This means that the diameter of the circular cross-section will be the same as the diagonal of one of the faces of the cube.
   - For a cube with side length \( s \), the diagonal of one face (which is a square) can be calculated using the Pythagorean theorem:
     \[
     \text{Diagonal of a square face} = s\sqrt{2}
     \]
   - The diameter of the circular cross-section is twice the radius, which is \( 2 \times 12.7 = 25.4 \, \text{cm} \).
   
2. **Relating Cube Dimensions to the Circle:**
   - Since the diagonal of the square face is equal to the diameter of the circle, we set:
     \[
     s\sqrt{2} = 25.4
     \]
     Solving for \( s \):
     \[
     s = \frac{25.4}{\sqrt{2}} = \frac{25.4}{1.414} \approx 17.95 \, \text{cm}
     \]
   
3. **Finding the Volume of the Cube:**
   - The volume \( V \) of a cube is given by:
     \[
     V = s^3 = (17.95)^3 \approx 5783.33 \, \text{cm}^3
     \]
   This is close to the answer provided, \( 5793.70 \, \text{cm}^3 \), accounting for possible rounding differences.

4. **Finding the Surface Area of the Cube:**
   - The surface area \( T \) of a cube is given by:
     \[
     T = 6s^2 = 6(17.95)^2 \approx 6 \times 322.41 \approx 1934.46 \, \text{cm}^2
     \]
   This is very close to the provided answer \( 1935.48 \, \text{cm}^2 \).

Thus, the largest cube that can be cut from the log has:
- Volume \( \approx 5793.70 \, \text{cm}^3 \)
- Total surface area \( \approx 1935.48 \, \text{cm}^2 \)

Would you like more details or have any questions?

### Related Questions:
1. How do we derive the diagonal formula for a cube’s face?
2. What happens if the shape inside the cylinder is not a cube, but a rectangular prism?
3. How do we calculate the volume of a cylinder instead of a cube?
4. What would be the size of the largest sphere that can fit inside the same log?
5. How can we generalize this approach for any polygon inside a circular cross-section?

### Tip:
Always remember that for a cube, the diagonal of the square face plays a key role in problems involving inscribed shapes in a circular cross-section.

Find the volume and total area of the largest cube of wood that can be cut from a log of circular cross-section whose radius is 12.7 cm.

Learn how to calculate the volume and surface area of the largest cube that can be cut from a circular log with a radius of 12.7 cm. Using key geometric principles, this problem applies the Pythagorean theorem and cube volume formulas. Ideal for Grades 10-12.

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