GATE CS 2024 Set 1 — Question 7

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Last updated 20 August 2026

Question

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

Correct answer

(A) 1/π

Solution

1.Calculate the dimensions of the cylinder:

The sheet has dimensions $54 \text{ cm} \times 4 \text{ cm}$ .
The problem states that the two longer edges are joined together. The longer edges have length $54 \text{ cm}$ . Joining them creates a seam of length $54 \text{ cm}$ , which becomes the height ( $h$ ) of the cylinder.
The other dimension ( $4 \text{ cm}$ ) becomes the circumference ( $C$ ) of the base of the cylinder.
$h = 54 \text{ cm}$
$C = 2\pi r = 4 \implies r = \frac{4}{2\pi} = \frac{2}{\pi} \text{ cm}$
Volume of the cylinder ( $V_{cyl}$ ) = $\pi r^2 h = \pi \left(\frac{2}{\pi}\right)^2 \times 54 = \pi \left(\frac{4}{\pi^2}\right) \times 54 = \frac{216}{\pi} \text{ cm}^3$

2. Calculate the dimensions of the cube:

The surface area of the cube is equal to the area of the sheet.
Area of sheet = $54 \times 4 = 216 \text{ cm}^2$ .
Surface area of a cube = $6a^2$ , where $a$ is the side length.
$6a^2 = 216 \implies a^2 = 36 \implies a = 6 \text{ cm}$ .
Volume of the cube ( $V_{cube}$ ) = $a^3 = 6^3 = 216 \text{ cm}^3$ .

3. Calculate the ratio:

Ratio = $\frac{V_{cyl}}{V_{cube}} = \frac{216/\pi}{216} = \frac{1}{\pi}$ .

GATE CS 2024 Set 1 — Question 7

Question

Correct answer

Solution

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