Directions: Read the information carefully and answer the following questions.…

2023

Directions: Read the information carefully and answer the following questions.
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 √3R cm and height H cm has volume 4950 cm3.

Find the ratio of the total surface area of the sphere to the lateral surface area of the cube.

  1. A.

    33: 17

  2. B.

    33: 19

  3. C.

    31: 17

  4. D.

    33: 14

  5. E.

    29: 17

Show answer & explanation

Correct answer: D

Concept: When a cube is inscribed in a sphere so that the sphere passes through every vertex, the cube's space diagonal equals the sphere's diameter. For a cube of side P this space diagonal is P√3, so P√3 = 2R. The total surface area of a sphere is 4πR2, and the lateral (curved) surface area of a cube is the area of its 4 side faces, 4P2.

Application:

  1. From the inscribed condition P√3 = 2R, square both sides: 3P2 = 4R2, so P2 = 4R2/3.

  2. Required ratio = (TSA of sphere) : (LSA of cube) = 4πR2 : 4P2 = πR2 : P2.

  3. Substitute P2 = 4R2/3: ratio = πR2 ÷ (4R2/3) = 3π/4. The R2 cancels, so the ratio does not depend on the actual size.

  4. Take π = 22/7: ratio = 3 × (22/7) ÷ 4 = 66/28 = 33/14, i.e. 33 : 14.

Note: Since the options are exact ratios of whole numbers, take π = 22/7 (the standard convention for such questions); the exact ratio 3π/4 then becomes 33/14.

Cross-check: 33 : 14 ≈ 2.357, which matches 3π/4 ≈ 2.356 — consistent. The cone (radius √3R, volume 4950 cm3) is extra data meant for other parts of this question set and is not needed for the surface-area ratio.

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