Folosim urmatoarele formule:
[tex]A_b=\frac{l^2\sqrt{3} }{4} \\\\A_B=\frac{L^2\sqrt{3} }{4} \\\\A_l=\frac{(P_b+P_B)\times a_t}{2}[/tex]
[tex]P_b=3l\\\\P_B=3L[/tex]
[tex]V=\frac{h}{3}(A_B+A_b+\sqrt{A_B\times A_b})[/tex]
[tex]a_b=\frac{l\sqrt{3} }{6} \\\\a_B=\frac{L\sqrt{3} }{6}[/tex]
Cunoastem:
l=3 cm
L=9 cm
apotema piramida (SE)=6 cm
a)
[tex]P_b=3\times 3=9\ cm\\\\P_B=3\times 9=27\ cm[/tex]
Notam trunchiul conform desen atasat
Calculam apotemele bazelor, OE si O'E'
[tex]OE=\frac{9\sqrt{3} }{6} =\frac{3\sqrt{3} }{2} \\\\O'E'=\frac{3\sqrt{3} }{6} =\frac{\sqrt{3} }{2}[/tex]
In ΔSOE avem OE║O'E'⇒
[tex]\frac{SE'}{SE} =\frac{SO'}{SO} =\frac{O'E'}{OE}[/tex]
Luam primul si ultimul raport, unde SE=6 cm
[tex]\frac{SE'}{6} =\frac{\frac{\sqrt{3} }{2} }{\frac{3\sqrt{3} }{2} } \\\\\frac{SE'}{6}=\frac{1}{3}[/tex]
SE'=2 cm
SE=SE'+EE'
EE'=6-2=4 cm (apotema trunchi)
[tex]A_l=\frac{(9+27)\times 4}{2} =72\ cm^2[/tex]
b)
[tex]A_b=\frac{9\sqrt{3} }{4}\ cm^2[/tex]
[tex]A_B=\frac{81\sqrt{3} }{4}[/tex]
Stim de la punctul anterior ca:
[tex]\frac{SE'}{SE} =\frac{SO'}{SO} =\frac{O'E'}{OE}\\\\\frac{SO'}{SO} =\frac{1}{3}[/tex]
- Luam trapezul O'E'EO si aplicam Pitagora (Suma catetelor la patrat este egala cu ipotenuza la patrat) in triunghiul E'FE, unde E'F⊥EO
FO=E'O'=[tex]\frac{\sqrt{3} }{2}[/tex]
[tex]EF=EO-EF=\frac{3\sqrt{3} }{2} -\frac{\sqrt{3} }{2} =\sqrt{3}[/tex]
EE'²=E'F²+EF²
16=E'F²+3
E'F=√13 cm=O'O
[tex]\frac{SO'}{SO} =\frac{1}{3}[/tex]
Facem proportie derivata si obtinem:
[tex]\frac{SO-SO'}{SO} =\frac{3-1}{3} \\\\\frac{\sqrt{13} }{SO} =\frac{2}{3} \\\\SO=\frac{3\sqrt{13} }{2}[/tex]
Calculam volum trunchi de piramida
[tex]V=\frac{\sqrt{13} }{3}( \frac{9\sqrt{3} }{4} +\frac{81\sqrt{3} }{4}+\sqrt{\frac{81\sqrt{3} }{4}\times \frac{9\sqrt{3} }{4} }=\frac{\sqrt{13} }{4} (\frac{90\sqrt{3} }{4} +\frac{27\sqrt{3} }{4})= \frac{\sqrt{13} }{4}\times \frac{117\sqrt{3} }{4} =\frac{117\sqrt{39} }{16} \ cm^3[/tex]
Calculam volum piramida
[tex]V=\frac{A_B\times SO}{3}[/tex]
[tex]V=\frac{\frac{81\sqrt{3} }{4}\times \frac{3\sqrt{13} }{2} }{3}[/tex]
[tex]V=\frac{81\sqrt{39} }{8} \ cm^3[/tex]
