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GATE 2024 CE (CE2) – Question 7

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

Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder.

[Figure: a circle containing seven equal circles, one in the centre and six around it, each touching the centre one, its two neighbours and the container.]

The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is

Diagram for GATE 2024 CE (CE2) question 7
  1. 5/2
  2. 7/2
  3. 9/2
  4. 3

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Show answer and explanation

Correct answer: (B) 7/2

Explanation

If the chalk radius is $r$, the container radius is $3r$ (the central circle plus the diameter of an outer circle on each side). The container's cross-section is $9\pi r^2$ and the chalk occupies $7\pi r^2$, so the empty part is $2\pi r^2$. The ratio is $\frac{7}{2}$.