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You are given marbles of diameter 10mm.They are to be placed such their centres 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 maximum number of marbles can be placed inside the area?Derive the expression for the number of molecules per unit area.

$\begin{array}{1 1}1.125\\2.125\\3.125\\4.125\end{array} $

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CD=$10\sin 60^{\large\circ}=10\times \large\frac{\sqrt 3}{2}$
$\Rightarrow 5\sqrt 3$
Area of square having spherical marbles in it.
$\Rightarrow 40\times 10^{-1}\times 40\times 10^{-1}cm^2$
$\Rightarrow 16cm^2$
The maximum no. of spheres of 10mm diameter in hcp packing can be seen.
Total length covered by spheres =$5+4\times CD$
$\qquad\qquad\qquad\qquad\quad\quad\;\;\;=5+4\times 10\sin 60^{\large\circ}$
$\qquad\qquad\qquad\qquad\quad\quad\;\;\;=5+4\times 5\sqrt 3=40$ mm
Maximum number of spheres =14(full) + 8(half) = 18
$\therefore$ Number of spheres per $cm^2=\large\frac{18}{16}$$=1.125$
answered Aug 1, 2014 by sreemathi.v

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