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

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1:1

5:4

7:5

3:2

answer is D.

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Detailed Solution

Given that the length, breadth and height of a rectangular parallelepiped are in the ratio 6:3:1
Let

l = 6 x, b = 3 x, h = x

We know that
Total surface area of parallelepiped

⇒ lateral surface area + 2 base area

⇒ perimeter of base × height + length × height ⇒ 2 (l + b) × h + 2 × l × b

⇒ 2 (6 x + 3 x) × x + 2 × 6 x × 3 x ⇒ 2 (9 x) × x + 2 × 6 x × 3 x → (1)

Now total surface area of cube

= 6 × Side 2 → (2)

But it is given that the surface areas of both the diagrams are the same. Thus from 1 and 2,

⇒ 2 (9 x) × x + 2 × 6 x × 3 x = 6 × Side 2

Dividing the above equation by 2,

⇒ (9 x) × x + 6 x × 3 x = 3 × Side 2

⇒ 9 x 2 + 18 x 2 = 3 × Side 2 ⇒ 27 x 2 = 3 × Side 2

Dividing both sides by 3,

⇒ 9 x 2 = Side 2

Taking square root on both sides

⇒ 3 x = s i d e

Now let’s go towards the volume.

V (c u b e) V (p a r a l l e l e p i p e d) = s i d e 3 l × b × h

Substituting the values

V (c u b e) V (p a r a l l e l e p i p e d) = (3 x) 3 6 x × 3 x × x V (c u b e) V (p a r a l l e l e p i p e d) = 27 x 3 18 x 3 V (c u b e) V (p a r a l l e l e p i p e d) = 32

Hence, the correct option is 4.

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