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? What expression represents the volume of the platform?

Answer 1

To find the surface area and the volume of the platform given the expressions for its faces, we first need to clarify what Face A, Face B, and Face C represent. It seems these might correspond to the areas of different faces of a three-dimensional object, not just a "platform."

For simplicity, let's assume:

- **Face A** has an area given by \(A_1 = 3x^2 + x\).
- **Face B** has an area given by \(A_2 = x^2 + x\).
- **Face C** has an area given by \(A_3 = 3x^2 + 4x + 1\).

### Surface Area Calculation

The surface area \(S\) of the platform, including the bottom, would be the sum of the areas of all faces involved. Assuming Face A, Face B, and Face C form the sides or top, we also need to include the bottom face, which we can assume has the same area as one of the previous faces (this is a common assumption, but it may not hold for all configurations):

If we take the bottom face to be similar to Face A, the total surface area would be:

\[
S = A_1 + A_2 + A_3 + A_1 \text{ (for the bottom face)}
\]

Thus,

\[
S = (3x^2 + x) + (x^2 + x) + (3x^2 + 4x + 1) + (3x^2 + x)
\]

Now, combine the terms:

\[
S = (3x^2 + 3x^2 + 3x^2 + x^2) + (x + x + 4x + x) + 1
\]

Combine like terms:

\[
S = (3 + 3 + 3 + 1)x^2 + (1 + 1 + 4 + 1)x + 1
\]

\[
S = 10x^2 + 7x + 1
\]

Thus, the expression for the surface area \(S\) of the platform is:

\[
S = 10x^2 + 7x + 1
\]

### Volume Calculation

Now, to find the volume \(V\) of the platform, we need to know the height of the platform and how it relates to the faces given. Without specific height information or a detailed shape (like whether it's a prism, cuboid, or other shape), we cannot calculate the volume directly.

If we assume a height \(h\) related to the dimensions of one of the faces, and assuming the volume is dictated by one of the areas (say, Face C), then:

\[
V = A \cdot h
\]
However, without further information on height or the relationship between the faces, an explicit volume can't be calculated.

To summarize:
- The expression for the surface area is \(S = 10x^2 + 7x + 1\).
- The volume expression depends on additional information (like height \(h\)) and the specific geometric context. For a simple estimation, one might use \(V = (3x^2 + 4x + 1)h\) given it could be taken as the base area with height \(h\).