A, B, C denote the areas of three co-terminus faces of a cuboid. If P and S denote the product and sum of dimensions of the cuboid respectively, which one of the following is correct ?

Options

A. `"PS"=A^(2)+B^(2)+C^(2)`
B. `"PS"=AB+BC+CA`
C. `P=S(A+B+C)`
D. `"SP"=(A+B+C)^(2)`

Answer

To solve the problem, we need to understand the relationship between the areas of the coterminous faces of a cuboid and its dimensions. Let's denote the dimensions of the cuboid as follows:

- Length = L
- Breadth = B
- Height = H

The areas of the three coterminous faces of the cuboid can be expressed as:

1. Area A (Length x Breadth) = L * B
2. Area B (Breadth x Height) = B * H
3. Area C (Height x Length) = H * L

Given:
- P = L * B * H (the product of the dimensions)
- S = L + B + H (the sum of the dimensions)

Now, we need to find the relationship between P and S in terms of the areas A, B, and C.

### Step-by-step Solution:

1. **Identify the areas of the coterminous faces:**
- A = L * B
- B = B * H
- C = H * L

2. **Express P and S:**
- P = L * B * H
- S = L + B + H

3. **Calculate P * S:**
- P * S = (L * B * H) * (L + B + H)

4. **Expand the expression:**
- P * S = L * B * H * (L + B + H)
- = L * B * H * L + L * B * H * B + L * B * H * H
- = L^2 * B * H + L * B^2 * H + L * B * H^2

5. **Relate this to the areas A, B, and C:**
- Notice that:
- L^2 * B * H = A * H
- L * B^2 * H = B * L
- L * B * H^2 = C * B
- Therefore, we can express the product as:
- P * S = A * H + B * L + C * B

6. **Final Expression:**
- The relationship derived shows that the product of the dimensions and the sum of the dimensions is related to the areas of the coterminous faces.

### Conclusion:
From the derived relationship, we can conclude that the correct option relates to the expression involving the areas of the coterminous faces.