Explanation:

To find the volume of the cuboid, let's break down the problem:

- Dimensions Ratio:

The ratio of the length, width, and height is 6:3:2. Let the common factor be \( x \).

So, the length \( l = 6x \), width \( w = 3x \), and height \( h = 2x \).

- Total Surface Area Formula:

The surface area \( A \) of a cuboid is given by \( A = 2(lw + lh + wh) \).

- Given Surface Area:

The total surface area is given as 1800 cm².

- Equation Setup:

\( 1800 = 2((6x)(3x) + (6x)(2x) + (3x)(2x)) \).

- Equation Solution:

Simplify to find \( x \):

$$

1800 = 2(18x^2 + 12x^2 + 6x^2) = 2(36x^2) = 72x^2

$$

$$

1800 = 72x^2

$$

$$

x^2 = \frac{1800}{72} = 25

$$

$$

x = 5

$$

- Calculate Dimensions:

Length \( l = 6 \times 5 = 30 \) cm

Width \( w = 3 \times 5 = 15 \) cm

Height \( h = 2 \times 5 = 10 \) cm

- Calculate Volume:

Volume \( V = l \times w \times h = 30 \times 15 \times 10 = 4500 \) cm³

- Highlight Correct Option:

Option 3: 4500 cm³