Let us consider the situation where the largest possible cube (shaded in grey) fits perfectly in a particular cylinder as shown in the diagram below; however, the question does not state that we must fit in a largest possible cube into the cylinder.

Let the edge of the cube be \(a\).

Since the cube has to have a perfect fit (minimum non-utilization of cylinder space),

we must have:

Height of the cylinder = edge of the cube = \(a\)

In right angled triangle CAB:

\(CB^2\) \(=\) \(CA^2\)+\(AB^2\) \(=\) \(a^2\) + \(a^2\) \(=\) \(2a^2\)

\(=> CB =a\sqrt{2}\)

Thus, the diameter of the cylinder \(= a\sqrt{2}\)

=> Radius of the cylinder \(=\)\(\frac{a}{\sqrt{2}}\)

Thus, volume of the cylinder

\(=\) \(\pi\) X \(radius^2\) X \(height\)

\(=\) \(\pi\) X \((\frac{a}{\sqrt{2}})^2\) X \(a\)

\(= \frac{3}{2}a^3\)

Volume of the Cube \(= a^3\)

Thus, volume of the cylinder not occupied by the cube

\(=\) \(\frac{3}{2}\)\(a^3\)\(-\)\(a^3\)

\(= \frac{a^3}{2}\)

Thus, the required percent

\(= (\frac{a^3}{2})/(\frac{3}{2}a^3) X 100\)

\(=\) 33.3 %

The above situation depicts the case where minimum volume of the cylinder as a percent of the

total volume of the cylinder is unutilized, i.e. not covered by the cube.

Thus, in any other scenario, the required percent value would be either greater than or equal to

33.3%

The only possible value from the answer options is 36%

The correct answer is option E.

Attachments

Cylinder.PNG [ 9.5 KiB | Viewed 455 times ]