Question

The surface area of one prism is
  620 in. squared
stacking four identical prisms on top of each other​ end-to-end so their smallest faces overlap.
Then find the total surface area of the exhibit.

Answer 1

To find the total surface area of four identical prisms stacked on top of each other, we first need to understand how stacking them affects the surface area.

1. **Surface Area of One Prism**: Given that the surface area of one prism is 620 in².

2. **Stacking Prisms**: When we stack the prisms, the surface areas where they touch each other will not be counted in the total surface area. Each prism has two small faces (the ends) that will be touching each other when they are stacked.

3. **Calculating Overlap**: When stacking four prisms, the two small faces of the top prism, the two small faces of the bottom prism, and the small faces shared between adjacent prisms (3 overlaps in total for the four prisms) are not accounted for in the total surface area.

- Let’s denote the area of the smallest face (the two surfaces that are touching when stacked) as \( A_s \).

4. **Original Surface Area Calculation**:
- Total surface area of four identical prisms without considering the stacking: \( 4 \times 620 \, \text{in}^2 = 2480 \, \text{in}^2 \).

5. **Identifying the Area of Overlapping Faces**: We could find the area of one of the small faces \( A_s \) if we had additional information about the dimensions of the prism. However, since we do not have the specific dimensions, let’s analyze how the overlaps affect the total.

- For four prisms:
- 1 face from the bottom prism
- 1 face from the top prism
- 3 interfaces between the prisms

Thus, we need to subtract these overlapping areas from the total.

6. **Total Surface Area Calculation**:
\[
\text{Total Surface Area} = (4 \times 620) - (3 \cdot A_s)
\]

Without knowing \( A_s \), we can’t provide an exact numerical answer, but we can clearly outline the formula:
- If you know \( A_s \), substitute it into the equation above to find the total surface area.

Assuming \( A_s \) is calculated or given, you would finalize your total surface area by plugging it into the equation. If you have the dimensions or know the area of the small face, please provide that to complete the calculation.