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GATE 2024 AE – Question 7

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

A rectangular paper sheet of dimensions 54 cm × 4 cm is taken. The two longer edges of the sheet are joined together to create a cylindrical tube. A cube whose surface area is equal to the area of the sheet is also taken. Then, the ratio of the volume of the cylindrical tube to the volume of the cube is

  1. $1/\pi$
  2. $2/\pi$
  3. $3/\pi$
  4. $4/\pi$

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Correct answer: (A) $1/\pi$

Explanation

The sheet is $54\times4$ cm. Joining the two **longer** edges (the 54 cm edges) together rolls the sheet so that the 54 cm edge becomes the height $h$ of the tube and the 4 cm edge becomes the circumference.

**Cylinder:** $2\pi r=4\Rightarrow r=\dfrac{2}{\pi}$, $h=54$.
$$V_{cyl}=\pi r^2h=\pi\cdot\frac{4}{\pi^2}\cdot54=\frac{216}{\pi}$$

**Cube:** its surface area equals the sheet area $54\times4=216$. So $6a^2=216\Rightarrow a=6$ and $V_{cube}=a^3=216$.

**Ratio:** $\dfrac{216/\pi}{216}=\dfrac1\pi$, which is option (A).