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The problem involves finding the volume and total area of the largest cube that can be cut from a log of wood with a circular cross-section, where the radius of the log is 12.7 cm.

To solve this, we need to determine the side length of the largest cube that can fit within the diameter of the log. The side length of the cube, \( s \), will be equal to the diameter of the circle, as the cube must fit within this diameter.

1. **Determine the side length of the cube:**
   Since the radius of the log is 12.7 cm, the diameter is \( 2 \times 12.7 \) cm = 25.4 cm. Thus, \( s = 25.4 \) cm.

2. **Calculate the volume of the cube:**
   The volume \( V \) of a cube is given by \( s^3 \).
   \[
   V = (25.4 \, \text{cm})^3 = 16387.064 \, \text{cm}^3
   \]

3. **Calculate the surface area of the cube:**
   The total surface area \( A \) of a cube is \( 6s^2 \).
   \[
   A = 6 \times (25.4 \, \text{cm})^2 = 3875.76 \, \text{cm}^2
   \]

Let's perform these calculations for exactness.The volume of the largest cube that can be cut from the log is \( 16,387.064 \, \text{cm}^3 \) and the total surface area is \( 3,870.96 \, \text{cm}^2 \).

Would you like to explore more about this or any other questions?

1. How does the size of the cube compare to the original log?
2. Can you calculate how much of the log's volume is unused after the cube is cut out?
3. What would be the dimensions and volume of a cylinder that could be cut from the same log?
4. How would the results change if the log had a different radius?
5. Are there any practical considerations for cutting a cube from a cylindrical log?

**Tip:** When solving geometric problems involving objects within other objects, it's helpful to visualize their placements and use their symmetries to simplify calculations.

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 of circular cross-section with a radius of 12.7 cm. Understand key geometry concepts such as volume and surface area formulas, suitable for high school students in Grades 9-12.

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