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The problem asks to find the side length of a cube inscribed in a right circular cone, where the base of the cone has a radius of 4 and a height of 2. One face of the cube lies on the base of the cone.

Let's break down the situation:

1. The cone has:
   - **Base radius** \( r = 4 \)
   - **Height** \( h = 2 \)

2. The cube is inscribed such that one face lies in the base of the cone. This means that the diagonal of the base of the cube must fit within the radius of the cone's base.

### Step-by-step solution:

1. The diagonal of the square face of the cube must equal the diameter of the cone’s base, which is twice the radius of the cone’s base. The diagonal \( d \) of a square (which is the base of the cube) is related to the side length \( s \) by the formula:

   \[
   d = s\sqrt{2}
   \]

2. The diameter of the cone's base is \( 2r = 8 \), so the diagonal of the square must fit this:

   \[
   s\sqrt{2} = 8
   \]

3. Solving for \( s \):

   \[
   s = \frac{8}{\sqrt{2}} = 8 \times \frac{\sqrt{2}}{2} = 8 \times 0.7071 = 5.6569
   \]

Thus, the side length of the cube is approximately **5.66** (rounded to two decimal places).

Would you like further clarification, or to explore more details?

Here are some related questions:
1. How is the diagonal of a square derived in geometry?
2. How does inscribing a cube in different solids (cone, cylinder, sphere) affect the calculations?
3. What is the relationship between a cone's dimensions and an inscribed shape's dimensions?
4. How would the problem change if the height of the cone were different?
5. How can this problem be solved using calculus or optimization techniques?

**Tip:** When working with inscribed shapes, always pay attention to the geometric constraints imposed by the enclosing shape.

A right circular cone has a base of radius 4 and height 2. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side length of the cube?

Learn how to calculate the side length of a cube inscribed in a cone with a base radius of 4 and height of 2. This geometry problem involves key formulas such as the diagonal of a square and relationships between the cone's dimensions and the inscribed cube.

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