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. 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 A. 5/2 B. 7/2 C. 9/2 D. 3

GATE 2024 · General Aptitude · Geometry · medium

Answer: Occupied space : Empty space = 7/2

1. Determine container radius from packing geometry: Centre of outer circle is 2r from container axis (= r + r, centre-to-centre). Outer circle extends r more to the wall. So R = 2r + r = 3r.
2. Compute cross-sectional areas: Container area = pi*(3r)^2 = 9*pi*r^2. Occupied area (7 sticks) = 7*pi*r^2. Empty area = 9*pi*r^2 - 7*pi*r^2 = 2*pi*r^2.
3. Compute the required ratio: Ratio = 7*pi*r^2 / (2*pi*r^2) = 7/2