Question

What expression represents the surface area of the entire platform, including the bottom, if Face A = 3x^2 + x; Face B = x^2 + x; and Face C = 3x^2 + 4x + 1?

Answer 1

To find the surface area of the entire platform including the bottom, we need to identify how many faces there are and what surfaces they cover. Based on the data provided, we have three faces labeled A, B, and C. If we assume the platform is a three-dimensional shape (like a box or rectangular prism) with these expressions representing the dimensions of the individual faces, we can derive the overall surface area.

1. **Identifying Faces**: Let's denote:
- Face A = 3x² + x (possibly representing the area of one of the sides)
- Face B = x² + x (possibly representing another side area)
- Face C = 3x² + 4x + 1 (possibly representing the area of the other side)

Assuming the bottom is the only extra area (if it's not counted in A, B, or C), it might be similarly based on these expressions.

2. **Surface Area Calculation**:

Surface area \(S\) of a box made up of faces (A, B, C and the bottom) can generally be modeled as:
\[
S = 2(Area_{A} + Area_{B} + Area_{C}) + Area_{bottom}
\]

Assuming the bottom has a respective area which corresponds to either A, B, or C, you may choose any of them or derive a necessary expression based on one of these areas.

Therefore, a hypothetical calculation (assuming each face comes together):

- **Total Surface Area**:
\[
S = 2 \times (Face A + Face B + Face C)
\]
- Insertable format:
\[
S = 2 \times ((3x^2 + x) + (x^2 + x) + (3x^2 + 4x + 1))
\]
- Combine the terms:
\[
S = 2 \times (3x^2 + x^2 + 3x^2 + x + 4x + 1) = 2 \times (7x^2 + 6x + 1)
\]

3. **Final Result**:
Therefore, the expression that represents the surface area of the entire platform, including the bottom would be:
\[
S = 14x^2 + 12x + 2
\]

This gives the total surface area in terms of x, effectively covering all provided faces, assuming we are viewing this as a form of rectangular configuration where such expressions correspond to adequate areas.