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Directions (111-112): Read the information carefully and answer the following questions. A cube of side P cm is placed inside a sphere of radius R cm in such a way that the sphere touches all the vertices of the cube. A cone of radius \(\sqrt{3}R\) cm and height H cm has volume 4950 cm\(^3\). 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
5. 29: 17

Correct answer: 33: 17

Solution

For a cube inscribed in a sphere, the space diagonal equals the sphere’s diameter, so P\(\sqrt{3}\) = 2R. Using the cone volume gives a value of R that allows the sphere’s surface area and the cube’s lateral surface area to be expressed and compared, yielding 33:17.

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