The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm³ and the total surface area is 252 cm². The length of the longest edge is

Given that:

Volume of a block = 216cm³

and Total Surface area = 252 cm³

Breadth of a rectangular block = a

& Height of a rectangular block = ar

Since, they are in G.P so there common ratio is same.

and we know that,

Volume of a rectangular solid block = L × B × H

[given]

⇒ 216 = a³

⇒ a = 6 …(i)

Now,

Total Surface Area of a block = 2[L×B + B×H + H×L]

[given]

[from (i)]

⇒ 2(r² + r + 1) = 7r

⇒ 2r² + 2r + 2 – 7r = 0

⇒ 2r² – 5r + 2 = 0

⇒ 2r² – 4r – r + 2 = 0

⇒ 2r(r – 2) – 1(r – 2)= 0

⇒ (2r – 1)(r – 2) = 0

⇒ 2r – 1 = 0 & r – 2 = 0

and r = 2

∴ Three unequal edges of the given solid block are:

Case 1: If a = 6 and r = 2, then

Breadth = a = 6

and Height = ar = 6 × 2 = 12

Case 2:

Breadth = a = 6

and

So, the length of the longest side = 12units

Hence, the correct option is (a)