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UPSC CDS I 2019 Math Paper

Question Numbers: 91-93

Consider the following for the next three (03) items:

A cube is inscribed in a sphere. A right circular cylinder is within the cube touching all the vertical faces. A right circular cone is inside the cylinder. Their heights are same and the diameter of the cone is equal to that of the cylinder.

Question : 92 of 100

Marks: +1, -0

What is the ratio of the volume of the cube to that of the cylinder?

4 : 3
21 : 16
14 : 11
45 : 32

Validate

Solution:

Concept:

The ratio of volumes when a cylinder is inscribed in a cube (height and diameter equal to cube side).

Explanation:

Let the side of the cube be

a

Volume of cube =

a^3

For the cylinder inscribed: height

h = a

, radius

r = \frac{a}{2}

Volume of cylinder =

\pi r^2 h = \pi \left(\frac{a}{2}\right)^2 a = \frac{\pi a^3}{4}

Ratio (cube : cylinder) =

a^3 : \frac{\pi a^3}{4} = 1 : \frac{\pi}{4} = \frac{4}{\pi}

Using

\pi \approx \frac{22}{7}

\frac{4}{22/7} = \frac{4 \times 7}{22} = \frac{28}{22} = \frac{14}{11}

Thus the ratio is

14 : 11

Answer:

C. 14 : 11

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