Discussion
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
Answer: B
The ratio is 7/2.
Let each chalk-stick have radius \(r\). In the hexagonal packing, the outer sticks touch the central one, so their centres are \(2r\) from the axis and their outer edges \(3r\) from it. The container's radius is therefore \(3r\).
\[ \begin{aligned} \text{occupied} &= 7\pi r^{2} \\ \text{container} &= \pi(3r)^{2} = 9\pi r^{2} \\ \text{empty} &= 2\pi r^{2} \end{aligned} \]
The common length cancels, so occupied : empty = 7 : 2. 9/2 compares the container with the empty space, and 5/2 and 3 do not follow from the radius \(3r\).
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