The problem involves a rectangular parallelepiped with dimensions in the ratio of 6:3:1. Let's denote the length, breadth, and height as 6x, 3x, and x respectively.
Calculating Surface Area
The surface area (SA) of a rectangular parallelepiped is given by the formula:
SA = 2(lb + bh + hl)
Substituting the dimensions:
SA = 2(6x * 3x + 3x * x + x * 6x)
This simplifies to:
SA = 2(18x² + 3x² + 6x²) = 2(27x²) = 54x²
Surface Area of the Cube
Let the side length of the cube be 'a'. The surface area of the cube is:
SA = 6a²
Setting the surface areas equal gives:
6a² = 54x²
From this, we can solve for 'a²':
a² = 9x²
Thus, a = 3x.
Calculating Volume
The volume (V) of the rectangular parallelepiped is:
V = l * b * h = 6x * 3x * x = 18x³
The volume of the cube is:
V = a³ = (3x)³ = 27x³
Finding the Volume Ratio
Now, we can find the ratio of the volume of the cube to the volume of the parallelepiped:
Volume Ratio = Volume of Cube : Volume of Parallelepiped = 27x³ : 18x³
This simplifies to:
Volume Ratio = 27 : 18 = 3 : 2
Final Answer
The ratio of the volume of the cube to the volume of the parallelepiped is 3:2. Therefore, the correct option is D.
