If the areas of three adjacent faces of a rectangular block are in the ratio of 2 : 3 : 4 and its volume is 9000 cu.cm; then the length of the shortest side is :

Answer: Option B

Let lb = 2x, bh = 3x and lh = 4x
Then,
$$\begin{array}{rl}& \\ & 24x^3={\left(lbh\right)}^2=9000\times 9000\\ & \Rightarrow x^3=375\times 9000\\ & \Rightarrow x=150\end{array}$$
So, lb = 300, bh = 450, lh = 600 and lbh = 9000
$$\begin{array}{rl}& \therefore h=\frac{9000}{300}=30\\ & l=\frac{9000}{450}=20\&\\ & b=\frac{9000}{600}=15\end{array}$$
Hence, shortest side = 15 cm