If the areas of adjacent faces of a cuboid (rectangular prism) are in the ratio of 2 : 3 : 5 and its volume is $900\, cm^3$, then the length of the longest side is

15 cm

The correct answer is Option (3) → 15 cm

Let the sides of the cuboid be a, b, c.

Adjacent face areas are:

• $ab : bc : ca = 2 : 3 : 5$

So, let:

$ab = 2k,\quad bc = 3k,\quad ca = 5k$

Step 1: Find the volume

$\text{Volume} = abc$

Multiply the three equations:

$(ab)(bc)(ca) = (abc)^2 = (2k)(3k)(5k) = 30k^3$

$(abc)^2 = 30k^3$

Given volume $abc = 900$:

$900^2 = 30k^3$

$810000 = 30k^3$

$k^3 = 27000 \Rightarrow k = 30$

Step 2: Find the sides

$ab = 2k = 60,\quad bc = 3k = 90,\quad ca = 5k = 150$

Now:

$a = \sqrt{\frac{ab \cdot ac}{bc}} = \sqrt{\frac{60 \cdot 150}{90}} = \sqrt{100} = 10$

$b = \sqrt{\frac{ab \cdot bc}{ca}} = \sqrt{\frac{60 \cdot 90}{150}} = \sqrt{36} = 6$

$c = \sqrt{\frac{bc \cdot ca}{ab}} = \sqrt{\frac{90 \cdot 150}{60}} = \sqrt{225} = 15$

Step 3: Longest side

$15\ \text{cm}$