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The problem reads:

**"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? (Round your answer to two decimal places.)"**

### To solve this, let's break it down step by step:

1. **Cone dimensions**:
   - Base radius \( R = 4 \)
   - Height \( h = 2 \)

2. **Cube inscribed**:
   - The cube is inscribed such that one of its faces is contained in the base of the cone, and the top of the cube is located at some point along the height of the cone.

### Visualization:
- The cone's cross-section can be thought of as an isosceles triangle.
- The cube’s bottom face sits on the cone’s base, and the top of the cube is aligned with the sloping sides of the cone.
- The radius of the cone decreases linearly from the base to the tip of the cone.

### Important geometry:
- The key is to recognize that the side length of the cube, denoted by \( s \), relates to the changing radius of the cone as we move up from the base.

### Slope of the cone's slanted side:
The slope of the cone's side can be calculated by the ratio of the height to the radius:
\[
\text{slope} = \frac{h}{R} = \frac{2}{4} = \frac{1}{2}
\]

This slope tells us how much the radius decreases as we move up along the height of the cone.

### Finding the side length of the cube:
If we denote the height of the cube as \( s \) (which is also the cube's side length), then the cube fits into a region of the cone where the height is \( s \) and the corresponding radius of the cone at that height is \( 4 - 2s \) (because the radius linearly decreases as we move up the height).

At the top of the cube, the radius of the cone should equal half the side length of the cube (because the cube is symmetrical about the center of the cone):
\[
\frac{s}{2} = 4 - 2s
\]
Solving for \( s \):

Multiply both sides by 2:
\[
s = 8 - 4s
\]
Now add \( 4s \) to both sides:
\[
5s = 8
\]
Finally, divide by 5:
\[
s = \frac{8}{5} = 1.6
\]

### Conclusion:
The side length of the cube is **1.60 units**.

Would you like further details on this solution or clarifications?

### Related Questions:
1. How would the solution change if the cone's height were increased?
2. What if the cube were rotated or placed differently in the cone?
3. How can the volume of the cube be calculated from this side length?
4. What is the volume of the cone, given its dimensions?
5. How can this problem be extended to other shapes inscribed in a cone?

#### Tip:
When visualizing problems involving geometric shapes, drawing cross-sections and using symmetry can simplify the problem-solving process significantly.

A right circular cone has 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? (Round your answer to two decimal places.)

This problem involves finding the side length of a cube inscribed in a cone with base radius 4 and height 2. It covers important geometric and algebraic concepts, including cone geometry, slope calculations, and cube dimensions. Suitable for students in Grades 9-12.

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