Question

Um poliedro convexo só tem faces triangulares e quadrangulares. Ele Tem 12
arestas, e a soma dos ângulos internos de suas faces é 1800°. Quantas são suas
faces de cada tipo?

Answer 1

$\mathrm{Sendo\ \mathbf{x}\ o\ n\acute{u}mero\ de\ faces\ triangulares\ e\ \mathbf{y}\ o\ de\ quadrangulares.}$

$\textbf{Pela rela\c{c}\~ao das arestas, temos:}\\\\ \mathrm{A=\dfrac{n_1x+n_2y+\dots}{2}\ \to\ 12=\dfrac{3x+4y}{2}\ \to\ \boxed{\mathrm{3x+4y=24}}\ \mathbf{(I)}}$

$\textbf{Pela rela\c{c}\~ao de Euler, temos:}\\\\ \mathrm{V+F=A+2\ \to\ V+F=12+2\ \to\ V=14-F}\\\\ \mathrm{\Rightarrow F=x+y\ \to\ \boxed{\mathrm{V=14-(x+y)}}}$

$\mathbf{Pela\ soma\ dos\ \hat{a}ngulos\ internos,\ temos:}\\\\ \mathrm{S_i=(V-2).360\°\ \to\ 1800\°=(V-2).360\°}\\\\ \mathrm{14-(x+y)-2=\dfrac{1800\°}{360\°}\ \to\ 12-(x+y)=5}\\\\ \mathrm{x+y=12-5\ \to\ x+y=7\ \to\ \boxed{\mathrm{y=7-x}}\ \mathbf{(II)}}$

$\mathbf{Substituindo\ (II)\ em\ (I):}\\\\ \mathrm{3x+4y=24\ \to\ 3x+4(7-x)=24}\\\\ \mathrm{3x+28-4x=24\ \to\ 28-x=24}\\\\ \mathrm{x=28-24\ \to\ \boxed{\mathrm{x=4}}}$

$\mathbf{Substituindo\ x\ em\ (II):}\\\\ \mathrm{y=7-x\ \to\ y=7-4\ \to\ \boxed{\mathrm{y=3}}}$

$\mathrm{\mathbf{Resposta:}\ 4\ faces\ triangulares\ e\ 3\ quadrangulares.}$