| A parallelepiped whose sides are in ratio 2: 4: 8 have the same volume as a cube. The ratio of their surface area is [SSC (10+2) 2010] |
A) 7 : 5
B) 4 : 3
C) 8 : 5
D) 7 : 6
Correct Answer: D
Solution :
| Let the sides of the parallelepiped be 2x, 4x and 8x units, respectively and the edge of cube be a units. |
| According to the question, |
| \[2x\times 4x\times 8x={{a}^{3}}\] |
| \[\Rightarrow \] \[8\times 8{{x}^{3}}={{a}^{3}}\] |
| Taking cube roots, |
| \[4x=a\] ... (i) |
| Surface area of parallelepiped |
| \[=2\,\,(lb+bh+hl)\] |
| \[=2\,\,(2x\times 4x+4x\times 8x+8x\times 2x)\] |
| \[=2\,\,(8{{x}^{2}}+32{{x}^{2}}+16{{x}^{2}})\] |
| \[=112{{x}^{2}}\,\,\text{units}\] |
| Surface area of cube \[=6{{a}^{2}}\,\text{units}\] |
| \[\therefore \]Ratio of surface area of parallelepiped and cube |
| \[=\frac{112{{x}^{2}}}{6{{a}^{2}}}=\frac{112{{x}^{2}}}{6\times 16{{x}^{2}}}\] [from Eq. (i)] |
| \[=7/6\] |
| \[\therefore \] Required ratio = 7: 6 |
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