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Question

# Area of regular hexagon varies directly as the square of its side. A regular hexagon having side 4 cm. has area $24\sqrt{3}{\mathrm{cm}}^{2}$. Find the area of regular hexagon having side 6 cm.

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Solution

## $\mathrm{Let}A\mathrm{and}S\mathrm{denote}\mathrm{the}\mathrm{area}\mathrm{and}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{side}\mathrm{of}\mathrm{hexagon},\mathrm{respectively}.\phantom{\rule{0ex}{0ex}}Thus,wehave:\phantom{\rule{0ex}{0ex}}A\alpha {S}^{2}\phantom{\rule{0ex}{0ex}}\therefore A=k{S}^{2}\left(k\mathrm{is}\mathrm{constant}.\right)\phantom{\rule{0ex}{0ex}}24\sqrt{3}=k×{4}^{2}\phantom{\rule{0ex}{0ex}}⇒k=\frac{24\sqrt{3}}{16}\phantom{\rule{0ex}{0ex}}⇒k=\frac{3\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}\therefore A=\frac{3\sqrt{3}}{2}{S}^{2}\left(\mathrm{Equation}\mathrm{of}\mathrm{variation}\right)\phantom{\rule{0ex}{0ex}}⇒A=\frac{3\sqrt{3}}{2}×{6}^{2}\phantom{\rule{0ex}{0ex}}⇒A=54\sqrt{3}\phantom{\rule{0ex}{0ex}}$

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