Aplicam urmatoarele formule:
[tex]P_b=3l[/tex]
[tex]A_l=\frac{P_b\times a_p}{2}[/tex]
[tex]A_b=\frac{l^2\sqrt{3} }{4}[/tex]
[tex]A_t=A_l+A_b[/tex]
[tex]V=\frac{A_b\times h}{3}[/tex]
[tex]a_b=\frac{l\sqrt{3} }{6}[/tex]
[tex]a_p^2=h^2+a_b^2[/tex]
[tex]m^2=\frac{l^2}{3} +h^2[/tex]
[tex]m^2=\frac{l^2}{4}+a_p^2[/tex]
a)
l=6
h=3
P=18
[tex]A_b=\frac{36\sqrt{3} }{4} =9\sqrt{3} \ cm^2[/tex]
[tex]a_b=\frac{6\sqrt{3} }{6} =\sqrt{3}[/tex]
[tex]a_p^2=9+3=12\\\\a_p=2\sqrt{3}[/tex]
[tex]A_l=\frac{18\times 2\sqrt{3} }{2} =18\sqrt{3}[/tex]
[tex]A_t=9\sqrt{3} +18\sqrt{3} } =27\sqrt{3}[/tex]
[tex]m^2=12+9=21\\m=\sqrt{21}[/tex]
[tex]V=\frac{9\sqrt{3}\times 3 }{3} =9\sqrt{3}[/tex]
b)
l=2
m=√13
[tex]13=\frac{4}{3}+h^2\\\\ h=\frac{\sqrt{35} }{\sqrt{3} } =\frac{\sqrt{105} }{3}[/tex]
[tex]a_b=\frac{\sqrt{3} }{3}[/tex]
[tex]a_p^2=\frac{105}{9}+\frac{3}{9} =\frac{108}{9}=12\\\\ a_p=2\sqrt{3}[/tex]
[tex]P_b=6[/tex]
[tex]A_b=\frac{4\sqrt{3} }{4}=\sqrt{3}[/tex]
[tex]A_l=\frac{6\times 2\sqrt{3} }{2} =6\sqrt{3}[/tex]
[tex]A_t=7\sqrt{3}[/tex]
[tex]V=\frac{\sqrt{3} \times \sqrt{105} }{9} =\frac{\sqrt{35} }{3}[/tex]
c)
m=13
[tex]a_p=12[/tex]
[tex]169=\frac{l^2}{4}+144 \\\\l^2=100\\\\l=10[/tex]
[tex]P_b=30[/tex]
[tex]A_b=\frac{100\sqrt{3} }{4}=25\sqrt{3}[/tex]
[tex]A_l=\frac{30 \times 12}{2}=180[/tex]
[tex]A_t=25\sqrt{3} +180[/tex]
[tex]a_b=\frac{10\sqrt{3} }{6} =\frac{5\sqrt{3} }{3}[/tex]
[tex]144=h^2+\frac{75}{9} \\\\h^2=\frac{1221}{9} \\\\h=\frac{\sqrt{1221} }{3}[/tex]
[tex]V=\frac{25\sqrt{3}\times \frac{\sqrt{1221} }{3} }{3}=\frac{25\sqrt{407} }{3}[/tex]
d)
l=2√3
Al=9√3
[tex]P_b=6\sqrt{3}[/tex]
[tex]9\sqrt{3} =\frac{6\sqrt{3} \times a_p}{2}[/tex]
[tex]a_p=3[/tex]
[tex]a_b=\frac{6}{6} =1[/tex]
[tex]A_b=\frac{12\sqrt{3} }{4}=3\sqrt{3}[/tex]
[tex]A_t=12\sqrt{3}[/tex]
[tex]9=1+h^2\\\\h=2\sqrt{2}[/tex]
[tex]V=\frac{3\sqrt{3}\times 2\sqrt{2} }{3} =2\sqrt{6}[/tex]
[tex]m^2=4+8\\\\m=2\sqrt{3}[/tex]
e)
Al=75√3
At=100√3
[tex]A_b=100\sqrt{3}-75\sqrt{3} =25\sqrt{3} =\frac{l^2\sqrt{3} }{4} \\\\l=10[/tex]
[tex]P_b=30\\\\75\sqrt{3} =\frac{30\times a_p}{2} \\\\a_p=5\sqrt{3}[/tex]
[tex]m^2=25+75=100\\\\m=10[/tex]
[tex]100=\frac{100}{3}+h^2\\\\ 3h^2=200\\\\h=\frac{10\sqrt{2} }{\sqrt{3} }=\frac{10\sqrt{6} }{3}[/tex]
[tex]V=\frac{25\sqrt{3}\times \frac{10\sqrt{6} }{3} }{3}=\frac{250\sqrt{2} }{3}[/tex]
f)
h=4√6
V=144√2
[tex]144\sqrt{2} =\frac{A_b\times 4\sqrt{6} }{3} \\\\A_b=\frac{108}{\sqrt{3} }=36\sqrt{3}[/tex]
[tex]36\sqrt{3} =\frac{l^2\sqrt{3} }{4} \\\\l=12[/tex]
[tex]m^2=48+ 96=144\\\\m=12[/tex]
[tex]144=36+a_p^2\\\\a_p=6\sqrt{3}[/tex]
[tex]P_b=36[/tex]
[tex]A_l=\frac{36\times 6\sqrt{3} }{2} =108\sqrt{3}[/tex]
[tex]A_t=144\sqrt{3}[/tex]
