The length, breadth and height of a cuboid are in the ratio 6 : 5 : 3. If its total surface area is 504 cm², find its dimensions. Also, find the volume of the cuboid.

Given:

The length, breadth and height of a cuboid are in the ratio 6 : 5 : 3.

The total surface area of the cuboid = 504 cm².

Let the length, breadth and height of cuboid be 6a, 5a and 3a.

As we know, the total surface area of the cuboid = 2(l x b + b x h + h x l)

⇒ 2(6a x 5a + 5a x 3a + 3a x 6a) = 504

⇒ 2(30a² + 15a² + 18a²) = 504

⇒ 2 x 63a² = 504

⇒ 126a² = 504

⇒ a² = $\dfrac{504}{126}$

⇒ a² = 4

⇒ a = $\sqrt{4}$

⇒ a = 2

Thus, length of the cuboid = 6a = 6 x 2 cm = 12 cm

Breadth of the cuboid = 5a = 5 x 2 cm = 10 cm

Height of the cuboid = 3a = 3 x 2 cm = 6 cm

As we know that volume of cuboid = l x b x h

= 12 x 10 x 6 cm³

= 720 cm³

Hence, the dimensions of the cuboid are 12 cm , 10 cm and 6 cm and the volume is 720 cm³.