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Official PYQ · 2024Set 1 · Q7MCQ2 marks

GATE CS 2024 Set 1 Q7 — General Aptitude

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​​A rectangular paper sheet of dimensions 54 cm×4 cm54 \mathrm{~cm} \times 4 \mathrm{~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

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Answer: A

Step 1: Volume of the cylindrical tube

When the two longer edges (each of length 54 cm54\text{ cm}) of the 54 cm×4 cm54\text{ cm} \times 4\text{ cm} sheet are joined together, the height of the cylinder is h=54 cmh = 54\text{ cm} and the circumference of the circular base is 2πr=4 cm2\pi r = 4\text{ cm}.

Solving for the radius rr:

r=42π=2π cmr = \frac{4}{2\pi} = \frac{2}{\pi}\text{ cm}

The volume of the cylinder (VcylinderV_{\text{cylinder}}) is:

Vcylinder=πr2h=π(2π)2×54=4π×54=216π cm3V_{\text{cylinder}} = \pi r^2 h = \pi \left(\frac{2}{\pi}\right)^2 \times 54 = \frac{4}{\pi} \times 54 = \frac{216}{\pi}\text{ cm}^3

Step 2: Volume of the cube

The surface area of the cube with side length aa is equal to the area of the rectangular sheet:

6a2=54×4=216 cm26a^2 = 54 \times 4 = 216\text{ cm}^2a2=36  ⟹  a=6 cma^2 = 36 \implies a = 6\text{ cm}

The volume of the cube (VcubeV_{\text{cube}}) is:

Vcube=a3=63=216 cm3V_{\text{cube}} = a^3 = 6^3 = 216\text{ cm}^3

Step 3: Ratio of the volumes

Ratio=VcylinderVcube=216/π216=1π\text{Ratio} = \frac{V_{\text{cylinder}}}{V_{\text{cube}}} = \frac{216/\pi}{216} = \frac{1}{\pi}

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