Question

The length, breadth, and height of a rectangular solid are in the ratio 6 : 4 :3. If the total surface area is 1728 cm². Find its dimensions.

Answer 1

Let the length, breadth and height of the rectangular solid be 'a', 'b' and 'c' respectively.
∴ a : b : c = 6 : 4 : 3     ...(Given)
Let the common multiple be x
a = 6x cm
b = 4x cm
c = 3x cm
The total surface area of the cuboid
= 1728cm²                ...(Given)
2(lb + bh + hl) = 1728
2(ab + bc + ca) = 1728
2[(6x . 4x) + (4x . 3x) + (3x . 6x)] = 1728

24x² + 12x² + 18x² = `(1728)/(2)`
54x² = 864

x² = `(864)/(54)`
x² = 16
x = `sqrt(16)`
∴ x = 4
∴ a = 6x = 6 x 4 = 24cm
∴ b = 4x = 4 x 4 = 16cm
∴ c = 3x = 3 x 4 = 12cm
Hence, the dimensions of the rectangular solid are 24cm, 16cm and 12cm.