GATE 2021 BT – Question 6

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

Two rectangular sheets M and N are each 6 cm × 1 cm. M is rolled by joining its short edges to form a cylinder. N is cut into equal square patches and assembled into the largest closed cube. Assuming the cylinder ends are closed, its volume divided by the cube volume is

$\pi/2$
$3/\pi$
$9/\pi$
$3\pi$

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Correct answer: (C) $9/\pi$

Explanation

**Cylinder M (6 cm × 1 cm).** The short edges (1 cm) are joined, so the 6 cm side wraps around to form the circumference and the height is 1 cm.

Circumference $2\pi r=6\Rightarrow r=\dfrac3\pi$ cm.
Volume $V_c=\pi r^2h=\pi\cdot\dfrac{9}{\pi^2}\cdot1=\dfrac9\pi\ \text{cm}^3$.

**Cube from N (6 cm × 1 cm).** A cube has 6 faces. Cutting N into six $1\times1$ cm squares gives exactly six faces, so the largest cube has edge 1 cm and volume $1\ \text{cm}^3$.

**Ratio:**
$$\frac{V_c}{V_{cube}}=\frac{9/\pi}{1}=\frac9\pi\quad(\text{option C}).$$