The problem asks to find the volume of a rectangular parallelepiped (also known as a rectangular prism or cuboid) given the areas of three distinct faces: $35 \text{ cm}^2$, $20 \text{ cm}^2$, and $28 \text{ cm}^2$.

1. Problem Description

The problem asks to find the volume of a rectangular parallelepiped (also known as a rectangular prism or cuboid) given the areas of three distinct faces: $35 \text{ cm}^2$, $20 \text{ cm}^2$, and $28 \text{ cm}^2$.

2. Solution Steps

Let the lengths of the sides of the rectangular parallelepiped be $a$, $b$, and $c$. The areas of the three distinct faces are then $ab$, $bc$, and $ac$. We are given that $ab = 35$, $bc = 20$, and $ac = 28$. We want to find the volume $V = abc$.

We can multiply the three given equations:

$(ab)(bc)(ac) = (35)(20)(28)$

$a^2 b^2 c^2 = (abc)^2 = 35 \cdot 20 \cdot 28 = (5 \cdot 7)(4 \cdot 5)(4 \cdot 7) = 5^2 \cdot 7^2 \cdot 4^2 = (5 \cdot 7 \cdot 4)^2 = 140^2$.

Taking the square root of both sides, we get:

$abc = \sqrt{140^2} = 140$.

Therefore, the volume $V = abc = 140 \text{ cm}^3$.

Note that option C has unit $\text{cm}^2$ which is an unit of area. Option C is incorrect.

Also note that option E has unit $\text{cm}^3$ which is an unit of volume.

3. Final Answer

The volume of the rectangular parallelepiped is $140 \text{ cm}^3$.

Option E is the correct option.

Final Answer: E. 140 cm³