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.