Question

Aria totala a unei piramide hexagonale regulate cu 4 radical 3cm patrati ,iar aria laterala este egala cu 5 radical 2 supra la 2 cm patrati.sa se afle lungimea laturii bazei piramidei.

Answer 1

 

$\displaystyle\bf\\A_t=4\sqrt{3}~cm^2\\\\ A_l=\frac{5\sqrt{2}}{2}~cm^2\\\\ A_b=A_t-A_l=4\sqrt{3}-\frac{5\sqrt{2}}{2}~cm^2=\\\\\\\boxed{\bf~A_b=\frac{8\sqrt{3}-5\sqrt{2}}{2}~~cm^2}\\\\\textbf{Baza piramidei este un hexagon regulat.}\\\textbf{Formula de calcul a ariei unui hexagon regulat este:}\\\\A_H=6\times\frac{L^2\sqrt{3}}{4}=\frac{6L^2\sqrt{3}}{4}=\boxed{\bf\frac{3L^2\sqrt{3}}{2}}$

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$\displaystyle\bf\\Calculam~latura:\\\\\frac{3L^2\sqrt{3}}{2}=\frac{8\sqrt{3}-5\sqrt{2}}{2}\\\\3L^2\sqrt{3}=8\sqrt{3}-5\sqrt{2}\\\\L^2=\frac{8\sqrt{3}-5\sqrt{2}}{3\sqrt{3}}=\frac{8}{3}-\frac{5}{3}\sqrt{\frac{2}{3}}\\\\\\ \boxed{\bf~L=\sqrt{\frac{8}{3}-\frac{5}{3}\sqrt{\frac{2}{3}}}~~cm}$