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Question Number 156423 by cortano last updated on 11/Oct/21

Commented by john_santu last updated on 11/Oct/21

$$answer=\frac{81(\sqrt{4\sqrt{3}-3}-1)}{2}squnits$$

Commented by cortano last updated on 11/Oct/21

$$\mathrm{ok}\mathrm{both}\mathrm{sirs}$$

Answered by mr W last updated on 11/Oct/21

$$rectanglea\times bwithb=9$$$$\tan \alpha =\frac{a}{b}=\frac{a}{9}$$$$\tan (\alpha +30)=\frac{a+9}{9}$$$$\frac{\frac{a}{9}+\frac{1}{\sqrt{3}}}{1-\frac{a}{9\sqrt{3}}}=\frac{a+9}{9}$$$$\frac{9+\sqrt{3}a}{9\sqrt{3}-a}=\frac{a+9}{9}$$$$a^2+9a-81(\sqrt{3}-1)=0$$$$a=\frac{9(\sqrt{4\sqrt{3}-3}-1)}{2}$$$$\Rightarrow ab=\frac{81(\sqrt{4\sqrt{3}-3}-1)}{2}$$

Commented by mr W last updated on 11/Oct/21

$$no!$$$$\tan \alpha =\frac{a}{b}=\frac{a}{9}$$$$\tan (\alpha +30)=\frac{a+9}{b}=\frac{a+9}{9}$$

Commented by cortano last updated on 11/Oct/21

$$\mathrm{typo}\mathrm{sir}\mathrm{it}\mathrm{should}\mathrm{be}\tan (\alpha -30°)$$

Commented by cortano last updated on 11/Oct/21

$$\mathrm{owh}\mathrm{but}\mathrm{the}\mathrm{answer}\frac{81\sqrt{4\sqrt{3}-3}}{2}-\frac{81}{2}$$$$\mathrm{ok}.\mathrm{thank}\mathrm{you}$$

Commented by mr W last updated on 11/Oct/21

$$pleaserechecksir!igot$$$$\frac{81\sqrt{4\sqrt{3}-3}}{2}-\frac{81}{2}$$$$not$$$$\frac{81\sqrt{4\sqrt{3}-1}}{2}-\frac{81}{2}$$

Commented by cortano last updated on 11/Oct/21

$$\mathrm{yes}.\mathrm{i}\mathrm{meant}\mathrm{it}.\mathrm{typo}$$

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