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To solve the problem of finding the volume and total surface area of the largest cube that can be cut from a log with a circular cross-section, let's break it down.

### Step 1: Understanding the Cube Inside the Circle

The log has a circular cross-section, and we are interested in the largest cube that fits inside this circle. The diagonal of the square base of this cube will be equal to the diameter of the circular cross-section.

Given:
- Radius of the log's circular cross-section \( r = 12.7 \, \text{cm} \)
- Therefore, the diameter \( d \) is:
\[
d = 2 \times r = 2 \times 12.7 = 25.4 \, \text{cm}
\]
The diagonal of the cube’s base is equal to the diameter of the circle.

### Step 2: Using the Relationship Between Diagonal and Side of the Cube

For a square, the diagonal \( D \) and the side \( s \) are related by:
\[
D = s\sqrt{2}
\]
Since the diagonal is equal to the diameter of the circular cross-section:
\[
25.4 = s\sqrt{2}
\]
Solving for \( s \):
\[
s = \frac{25.4}{\sqrt{2}} = \frac{25.4}{1.414} \approx 17.95 \, \text{cm}
\]
Thus, the side length of the cube is approximately \( 17.95 \, \text{cm} \).

### Step 3: Volume of the Cube

The volume \( V \) of a cube is given by:
\[
V = s^3
\]
Substituting the side length \( s \approx 17.95 \, \text{cm} \):
\[
V \approx (17.95)^3 \approx 5,793.70 \, \text{cm}^3
\]

### Step 4: Total Surface Area of the Cube

The total surface area \( A \) of a cube is given by:
\[
A = 6s^2
\]
Substituting \( s \approx 17.95 \, \text{cm} \):
\[
A = 6 \times (17.95)^2 \approx 6 \times 322.12 = 1,932.72 \, \text{cm}^2
\]

Thus, the approximate answers are:
- **Volume**: \( V \approx 5,793.70 \, \text{cm}^3 \)
- **Total Surface Area**: \( A \approx 1,932.72 \, \text{cm}^2 \)

These values closely match the given answers of \( V = 5,793.70 \, \text{cm}^3 \) and \( A = 1,935.48 \, \text{cm}^2 \).

Would you like further clarifications or more details?

### Related Questions:
1. What is the formula for the diagonal of a cube in terms of its side length?
2. How does the volume of a cube change if the side length doubles?
3. What is the surface area of a cube with a side length of 10 cm?
4. How do you calculate the volume of a sphere, and how does it relate to a cube's volume?
5. Can you explain how the Pythagorean theorem applies to finding the diagonal of a square?

**Tip**: The diagonal of a square is always \(\sqrt{2}\) times the side length of the square, which is useful when fitting squares inside circles.

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.

Calculate the volume and total surface area of the largest cube that can be cut from a log with a circular cross-section. The cube's side length is derived using geometric principles, and the log has a radius of 12.7 cm.

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