[latex]a) V= frac{1}{3}*Pp*H \a=6cm\P= frac{a ^{2} sqrt{3} }{4} \Pp=frac{6 ^{2} sqrt{3} }{4} =9 sqrt{3} cm ^{2} \H^2= 10^2-(frac{1}{3}h)^2\ frac{1}{3} h= frac{1}{3} * frac{a sqrt{3} }{2} \frac{1}{3} h= frac{1}{3} * frac{6 sqrt{3} }{2}= frac{6 sqrt{3} }{6} = sqrt{6} cm\H^2=10^2-( sqrt{3} )^2=100-3=97\H= sqrt{97} cm\V= frac{1}{3} *9 sqrt{3} * sqrt{97} =3 sqrt{3} * sqrt{97}= 3sqrt{97*3} =3 sqrt{291}[/latex] [latex]b) H=5cm\a=9\ frac{1}{3} h= frac{9 sqrt{3} }{2} * frac{1}{3} = frac{3 sqrt{3} }{2} \Pp= frac{9^2 sqrt{3} }{4}= frac{81 sqrt{3} }{4} cm^2\Ps= frac{9* frac{3 sqrt{3} }{2} }{2} = frac{9* 3sqrt{3} }{4} = frac{27 sqrt{3} }{4} \Pc=frac{81 sqrt{3} }{4} +4*frac{27 sqrt{3} }{4} = 27 sqrt{3} +frac{81 sqrt{3} }{4}[/latex] [latex]c)H=8cm \4 sqrt{3} = frac{2}{3} h \h=6 sqrt{3}cm \h =8cm\6 sqrt{3} = frac{a sqrt{3} }{2} |*2\12 sqrt{3} =a sqrt{3} |: sqrt{3} \a=12cm\ Pp= frac{12^2 sqrt{3} }{4} = frac{144 sqrt{3} }{4} =36 sqrt{3} cm^2 \V= frac{36 sqrt{3}*8 }{3} =96 sqrt{3} cm^2[/latex]
