Between a cube and a right circular cylinder, does the cube have a

Call the side of the cube is \(a\), the volume of the cube is \(a^3\) and the lateral surface area of the cube is \(4a^2\)

Call the height and the radius of the cylinder are \(h\) and \(b\). The volume of the cylinder is \(\pi hb^2\) and the lateral surface area of the cylinder is \(2\pi bh\)

(1) Since we could know \(h=2b\), however we can't know the value of \(a\). Insufficient.

(2) We have \(a^3 = \pi hb^2\). We have to compare \(4a^2\) and \(2\pi bh\)

Since \(a^3 = \pi hb^2 \implies 2 \pi bh = \frac{2a^3}{b}\) . We now have to compare \(\frac{2a^3}{b}\) and \(4a^2\) or we have to compare \(\frac{a}{b}\) and 2. However, we have no information about \(\frac{a}{b}\), insufficient.

Combine (1) and (2):

From (1) have \(h=2b\).

From (2) have \(a^3 = \pi hb^2 = 2\pi b^3 \implies \frac{a}{b} = \sqrt[3]{2\pi}\)

Now, from given condition we have \(\pi = (\frac{3}{2})^3 \\
\implies 2\pi = 2 * \frac{27}{8} = \frac{27}{4} < 8 \\
\implies \sqrt[3]{2\pi} < 2 \)

From this, we could compare \(\frac{a}{b}\) and 2 or we could compare \(\frac{2a^3}{b}\) and \(4a^2\). Sufficient.

The answer is C