The length, breadth and height of a cuboid (rectangular solid) are 4 : 3 : 2.

(i) If its surface area is 2548 cm², find its volume.

(ii) If its volume is 3000 m³, find its surface area.

(i) It is given that the length, breadth and height of a cuboid are 4 : 3 : 2.

The surface area = 2,548 cm².

Let the length, breadth and height of a cuboid be 4a, 3a and 2a.

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

⇒ 2(4a x 3a + 3a x 2a + 2a x 4a) = 2,548

⇒ 2(12a² + 6a² + 8a²) = 2,548

⇒ 2 x 26a² = 2,548

⇒ 52a² = 2,548

⇒ a² = $\dfrac{2,548}{52}$

⇒ a² = 49

⇒ a = $\sqrt{49}$

⇒ a = 7 cm

Thus, length of the cuboid = 4a = 4 x 7 cm = 28 cm

Breadth of the cuboid = 3a = 3 x 7 cm = 21 cm

Height of the cuboid = 2a = 2 x 7 cm = 14 cm

Volume of the cube = l x b x h

= 28 x 21 x 14 cm³

= 8,232 cm³

Hence, the volume of the cube is 8,232 cm³.

(ii) The volume of the cube = 3,000 m³

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

⇒ 4a x 3a x 2a = 3,000

⇒ 24a³ = 3,000

⇒ a³ = $\dfrac{3,000}{24}$

⇒ a³ = 125

⇒ a = $\sqrt[3]{125}$

⇒ a = 5 m

Thus, length of the cuboid = 4a = 4 x 5 m = 20 m

Breadth of the cuboid = 3a = 3 x 5 m = 15 m

Height of the cuboid = 2a = 2 x 5 m = 10 m

The surface area of cuboid = 2(l x b + b x h + h x l)

= 2(20 x 15 + 15 x 10 + 10 x 20) m²

= 2(300 + 150 + 200) cm²

= 2 x 650 cm²

= 1,300 m²

Hence, the surface area of the cuboid is 1,300 m².