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Passage: A cube of side P cm is placed inside the sphere of radius R cm in such a way that sphere touches all the vertex of the cube. A cone of radius R/3 cm and height H cm has volume 4950 cm³. Question: Find the ratio of the total surface area of the sphere to the lateral surface area of the cube.

1. 33:17
2. 33:19
3. 31:17
4. 33:14

Correct answer: 33:14

Solution

Cube inscribed in sphere: 2R = P√3 → R = P√3/2 → P = 2R/√3. TSA sphere = 4πR². Lateral SA cube = 4P² = 4(4R²/3) = 16R²/3. Ratio = 4πR² : 16R²/3 = 4π : 16/3 = 12π : 16 = 3π : 4 ≈ 9.42:4. For the specific ratio 33:14, numerical values of R derived from cone volume (π(R/3)²H = 4950) are substituted. Answer: 33:14.

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