The length, the breadth and the height of a cuboid are in the ratio 5 : 3 : 2. If its volume is 240 cm³, find its dimensions. Also, find the total surface area of the cuboid.

Given:

The length, breadth and height of a cuboid are in the ratio 5 : 3 : 2.

The volume of the cuboid = 240 cm³.

Let the length, breadth and height of cuboid be 5a, 3a and 2a.

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

⇒ 5a x 3a x 2a = 240 cm³

⇒ 30a³ = 240

⇒ a³ = $\dfrac{240}{30}$

⇒ a³ = 8

⇒ a = $\sqrt[3]{8}$

⇒ a = 2 cm

Thus, the length of the cuboid = 5a = 5 x 2 = 10 cm

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

Height of the cuboid = 2a = 2 x 2 = 4 cm

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

= 2(10 x 6 + 6 x 4 + 4 x 10) cm²

= 2(60 + 24 + 40) cm²

= 2 x 124 cm²

= 248 cm²

Hence, the dimensions of the cuboid are 10 cm , 6 cm and 4 cm and the total surface area is 248 cm².