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The problem given is about a rectangular prism (box) with the ratio of its length, width, and height being \(5:3:2\). The area of the base (length \(\times\) width) of the box is given as \(135 \text{ cm}^2\). We need to determine the dimensions of the box.

Let's denote the length, width, and height of the box as \(5k\), \(3k\), and \(2k\) respectively, where \(k\) is a scaling factor.

Given the area of the base:
\[ (5k) \times (3k) = 135 \text{ cm}^2 \]
\[ 15k^2 = 135 \]

We can solve for \(k^2\) and subsequently for \(k\), then calculate the dimensions of the box:
\[ k^2 = \frac{135}{15} \]
\[ k^2 = 9 \]
\[ k = \sqrt{9} \]
\[ k = 3 \]

With \(k = 3\):
- Length = \(5k = 15\) cm
- Width = \(3k = 9\) cm
- Height = \(2k = 6\) cm

Thus, the dimensions of the box are:
- **Length: 15 cm**
- **Width: 9 cm**
- **Height: 6 cm**

Would you like more details on this or have any other questions? Here are some related questions you might consider:

1. How would you determine the volume of this box?
2. What happens if the ratio of dimensions is changed?
3. How would you solve this problem if only the volume and one dimension were given?
4. Can you derive a general formula for the dimensions based on the area of the base and the ratio?
5. What are the implications of using different units, such as meters instead of centimeters?

**Tip:** When given a problem involving ratios, it's often helpful to introduce a scaling factor, represented by \(k\), to easily manipulate and solve equations that involve proportions.

Perbandingan panjang, lebar dan tinggi suatu balok adalah 5 : 3 : 2. Jika luas alas balok tersebut 135 cm², tentukan panjang, lebar dan tinggi balok tersebut!

Learn how to solve for the dimensions of a rectangular prism where the ratio of its length, width, and height is 5:3:2, and the area of its base is given as 135 cm². A detailed step-by-step guide is provided. This problem covers key algebraic and geometric concepts, suitable for students in Grades 7-9.

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