Question

You are given marbles of diameter $10mm$. They are to be placed such that their centers are lying in a square bond by four lines each of length $40mm$. What will be the arrangement of marbles in a plane, so that, the maximum number of marbles can be placed inside the area? Sketch the diagram and derive the expression for the number of marbles per unit area.

Answer 1

In order to accomodate maximum number of spheres, there should be hcp (hexagonal closed packing).
Area of square having spherical marbles $=16c{m}^2$
Maximum number of spheres $=14\left(full\right)+8\left(half\right)$
Number of spheres $=18$
Per unit area (per $c{m}^2)=\frac{18}{16}=1.125$
Length $PQ$ of square $=4cm$
Length $PR=5+4\times 5\sqrt{3}=40mm=4cm$
$CD=10\sin {60}^{\circ }=\frac{10\sqrt{3}}{2}=5\sqrt{3}$.