The length, breadth and height of a rectangular solid are in the ratio 5 : 4 : 2. If the total surface area is 1216 cm², find the length, the breadth and the height of the solid.

Given:

Total surface area = 1216 cm²

Ratio of dimensions = 5 : 4 : 2

Let the sides of rectangular solid be 5a, 4a and 2a.

Total surface area = 2(lb + bh + hl)

⇒ 2(5a x 4a + 4a x 2a + 2a x 5a) = 1216 cm²

⇒ 2(20a² + 8a² + 10a²) = 1216 cm²

⇒ 2 x 38a² = 1216 cm²

⇒ 76a² = 1216 cm²

⇒ a² = $\dfrac{1216}{76}$ cm²

⇒ a² = 16 cm²

⇒ a = $\sqrt{16}$ cm

⇒ a = 4 cm

Dimensions of rectangular solid are:

Length = 5a = 5 x 4 cm = 20 cm

Breadth = 4a = 4 x 4 cm = 16 cm

Height = 2a = 2 x 4 cm = 8 cm

Hence, length, breadth and height of rectangular solid are 20 cm, 16 cm and 8 cm, respectively.