GATE CS 2024 Set 1 Q.7: Quantitative Aptitude

A rectangular paper sheet of dimensions $54\text{ cm} \times 4\text{ cm}$ 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

$1/\pi$

$2/\pi$

$3/\pi$

$4/\pi$

Official answer and solution

Official answer: A
From: Official GATE 2024 answer key

Solution

Find the radius and height of the cylinder formed by joining the 54 cm edges, compute the side length of the cube from its surface area, and take the ratio of their volumes.

Volume of the cylindrical tube:
- Since the two longer edges ( $54\text{ cm}$ ) are joined together, the height of the cylinder is $h = 54\text{ cm}$ and the circumference of its circular base is the shorter edge $2\pi r = 4\text{ cm}$ , giving $r = \frac{2}{\pi}\text{ cm}$ .
- Volume of the cylinder: $V_{\text{cylinder}} = \pi r^2 h = \pi \left(\frac{2}{\pi}\right)^2 \times 54 = \frac{216}{\pi}\text{ cm}^3$
Volume of the cube:
- Area of the sheet $= 54 \times 4 = 216\text{ cm}^2$ .
- For a cube of side $a$ , total surface area $6a^2 = 216 \implies a^2 = 36 \implies a = 6\text{ cm}$ .
- Volume of the cube: $V_{\text{cube}} = a^3 = 6^3 = 216\text{ cm}^3$
Ratio of volumes: $\frac{V_{\text{cylinder}}}{V_{\text{cube}}} = \frac{216 / \pi}{216} = \frac{1}{\pi}$

Final answer

Option (A)

Common trap

Taking $2\pi r = 54\text{ cm}$ and $h = 4\text{ cm}$ (which corresponds to joining the two shorter edges instead of the two longer edges).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 7 of 65
Section: General Aptitude

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source