Question

ln a cuboid, the sum of the squares of its three dimensions is equal to half of its total surface area. If the volume of the cuboid is 729 cm³, then find its lateral surface area (in cm²).

Options

• 648

• 576

• 900

• 324

Answer 1

324

Explanation:

Let, dimensions of cuboid are `l`, b and h.

A.T.Q,

`l^2 + b^2 + h^2 = 1/2 xx 2(lb + bh + hl)`

⇒ `l^2 + b^2 + h^2 = lb + bh + hl`

⇒ `l^2 + b^2 + h^2 - lb - bh - hl` = 0

⇒ `1/2 (2l^2 + 2b^2 + 2h^2 - 2lb - 2bh - 2hl)` = 0

⇒ `(1 - b)^2 + (b - h)^2 + (h - l)^2` = 0

Therefore, `l = b = h`

Given, Volume of cuboid = 729 cm³

`l xx b xx h` = 729 cm³, `l^3` = 729 cm³

∴ `l` = 9 cm

Lateral surface area = `2(l + b) xx h`

= `2(l + l) xx l`

= 2(9 + 9) × 9

= 324