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The problem states that the dimensions of a rectangular parallelepiped are in the ratio 2:3:4, and the sum of all its edges equals 108 cm. We need to find the actual dimensions.

### Solution:
The sum of all the edges of a rectangular parallelepiped can be calculated using the formula:
\[
4(l + w + h) = 108,
\]
where \(l\), \(w\), and \(h\) are the lengths of the sides. 

### Step 1: Set up the equation
Let \(l = 2x\), \(w = 3x\), and \(h = 4x\), where \(x\) is a scaling factor.

### Step 2: Plug into the formula
\[
4(2x + 3x + 4x) = 108.
\]
This simplifies to:
\[
4 \cdot 9x = 108.
\]
Therefore,
\[
36x = 108.
\]
Solving for \(x\), we get:
\[
x = 3.
\]

### Step 3: Calculate the dimensions
- \(l = 2x = 2 \cdot 3 = 6\) cm,
- \(w = 3x = 3 \cdot 3 = 9\) cm,
- \(h = 4x = 4 \cdot 3 = 12\) cm.

Thus, the dimensions are \(6\) cm, \(9\) cm, and \(12\) cm.

The correct answer is:
**6 cm, 9 cm, 12 cm.**

Do you want further details or have any questions?

Here are five related questions:
1. How would the solution change if the sum of the edges was different?
2. What is the volume of the parallelepiped with dimensions 6 cm, 9 cm, and 12 cm?
3. How would you find the surface area of this parallelepiped?
4. If the ratio of the dimensions was 1:2:3 instead, what would the new dimensions be?
5. How can you verify that the dimensions obtained satisfy the given conditions?

**Tip:** For ratio-based problems, setting variables as multiples of a common factor helps simplify the calculations.

Знайдіть виміри прямокутного паралелепіпеда, якщо вони відносяться як 2:3:4, а сума всіх його ребер дорівнює 108 см.

Learn how to find the dimensions of a rectangular parallelepiped whose sides are in the ratio 2:3:4, given that the sum of all its edges equals 108 cm. This problem applies geometric formulas and basic algebra to solve for the dimensions.

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