In an 4 by 5 by 8 unit rectangular prism is painted green on all faces, then cut into cubes. There are 160 cubes. If both 4 by 5 of the block were painted red, and the remaining faces were painted green, how many of the unit cubes have at least one face that is painted green

To determine how many unit cubes have at least one face painted green, let's break down the problem step-by-step.

1. **Volume of the rectangular prism:**
   The dimensions of the rectangular prism are 4 units by 5 units by 8 units.
   \[
   \text{Volume} = 4 \times 5 \times 8 = 160 \text{ unit cubes}
   \]
   This confirms the total number of cubes is 160.

2. **Original coloring:**
   Initially, the entire prism is painted green on all six faces.

3. **Revised coloring:**
   The faces of the dimensions 4 by 5 are painted red, which are two opposite faces. The remaining faces (of dimensions 4 by 8 and 5 by 8) are painted green.

4. **Unit cubes with green faces:**
   Since the problem involves determining the unit cubes that have at least one green face after the red faces are painted, we need to count the unit cubes that touch any of the green faces.

5. **Identify unit cubes without any green face:**
   These are the unit cubes that are fully contained within the 4 by 5 faces (red faces). These unit cubes will not touch the green faces at all. The red faces are each one unit thick, so:
   - Each red face is a \(4 \times 5\) layer of cubes.

6. **Total unit cubes in the red faces:**
   Each of the \(4 \times 5\) faces contains:
   \[
   4 \times 5 = 20 \text{ unit cubes}
   \]
   Since there are two such faces:
   \[
   20 \times 2 = 40 \text{ unit cubes}
   \]

7. **Unit cubes touching green faces:**
   The total number of unit cubes in the prism is 160. We subtract the number of unit cubes that only touch the red faces:
   \[
   160 - 40 = 120 \text{ unit cubes}
   \]

Thus, there are **120 unit cubes** that have at least one face painted green.

Would you like more details or have any questions? Here are some additional questions you might find interesting:

1. How many unit cubes have exactly one face painted green?
2. How many unit cubes have exactly two faces painted green?
3. What is the surface area of the original rectangular prism?
4. If the dimensions were changed to 6 by 7 by 8, how many unit cubes would have at least one face painted green?
5. How many unit cubes have at least one face painted red?

**Tip:** When dealing with problems involving painted surfaces and cutting into smaller pieces, visualizing the layers and understanding the face contributions can simplify your calculations.

Explore how many unit cubes have at least one face painted green in a 4 by 5 by 8 unit rectangular prism. Learn the step-by-step solution involving volume calculations, surface area considerations, and cube distribution analysis. Suitable for Grades 6-8 students studying geometry and volume calculations.

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