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The length, breadth and height of a rectangular parallelopiped are in the ratios 6 : 3 : 1. If the surface area of a cube is equal to the surface area of this parallelopiped, then what is the ratio of the volume of the cube to the volume of the parallelopiped ? 

(a) 1 : 1 

(b) 5 : 4 

(c) 7 : 5 

(d) 3 : 2

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(d) 3 : 2

Let the length, breadth and height of the rectangular parallelopiped be 6\(x\), 3\(x\) and \(x\)

Let the side of the cube be a.

∴ By the given condition, 

Surface area of a cube 

= Surface area of rectangular parallelopiped 

6(a)2 = 2(6\(x\) x 3\(x\) + 3\(x\) x \(x\) + \(x\) x 6\(x\)

⇒ 6a2 = 2(18\(x\)2 + 3\(x\)2 + 6\(x\)2

⇒ 6a2 = 54\(x\)2 ⇒ a2 = 9\(x\)2 

⇒ a = 3\(x\) 

Now, Volume of cube : Volume of rectangular parallelopiped 

= a3 : (6\(x\) x 3\(x\) x \(x\)) = (3\(x\))3 : 18\(x\)3 = 27 : 18 = 3 : 2.

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