A parallelopiped whose sides are in ratio `2:4:8` have the same volume as a cube. The ratio of their surface area is :

Options

A. `7:5`
B. `4:3`
C. `8:5`
D. `7:6`

Answer

To solve the problem, we need to find the ratio of the surface area of a parallelepiped with sides in the ratio 2:4:8 to the surface area of a cube that has the same volume as the parallelepiped.

### Step-by-step Solution:

1. **Define the sides of the parallelepiped**:
Given the sides are in the ratio 2:4:8, we can express the sides as:
- Length = \(2x\)
- Width = \(4x\)
- Height = \(8x\)

2. **Calculate the volume of the parallelepiped**:
The volume \(V\) of a parallelepiped is given by the product of its sides:
\[ V = \text{Length} \times \text{Width} \times \text{Height} = (2x) \times (4x) \times (8x) = 64x^3 \]

3. **Define the side of the cube**:
Let the side of the cube be \(y\).

4. **Set the volumes equal**:
Since the volume of the cube is equal to the volume of the parallelepiped, we have:
\[ y^3 = 64x^3 \]

5. **Solve for \(y\)**:
Taking the cube root of both sides, we find:
\[ y = \sqrt[3]{64x^3} = 4x \]

6. **Calculate the surface area of the parallelepiped**:
The surface area \(S\) of a parallelepiped is given by:
\[ S = 2(\text{Length} \times \text{Width} + \text{Width} \times \text{Height} + \text{Height} \times \text{Length}) \]
Substituting the values:
\[ S = 2((2x)(4x) + (4x)(8x) + (8x)(2x)) \]
Calculating each term:
- \( (2x)(4x) = 8x^2 \)
- \( (4x)(8x) = 32x^2 \)
- \( (8x)(2x) = 16x^2 \)

Adding these:
\[ S = 2(8x^2 + 32x^2 + 16x^2) = 2(56x^2) = 112x^2 \]

7. **Calculate the surface area of the cube**:
The surface area \(S_c\) of a cube is given by:
\[ S_c = 6y^2 \]
Substituting \(y = 4x\):
\[ S_c = 6(4x)^2 = 6 \times 16x^2 = 96x^2 \]

8. **Find the ratio of the surface areas**:
The ratio of the surface area of the parallelepiped to the surface area of the cube is:
\[ \text{Ratio} = \frac{S}{S_c} = \frac{112x^2}{96x^2} = \frac{112}{96} = \frac{7}{6} \]

### Final Answer:
The ratio of the surface area of the parallelepiped to the surface area of the cube is \( \frac{7}{6} \).