Question

um prisma triangular regular de 225 m2 de area lateral tem a aresta da base igual a um terço da altura. calcule o volume desse prisma.

Answer 1

$\textrm{Dados do exerc\'icio:}\\\\ \textrm{P.T.R.}\ \to\ \textrm{prisma triangular regular}\\ \mathrm{A_{L}=225\ m^2\ \ \| \ \ a=\dfrac{h}{3}\ \to\ h=3.a}\\\\ \textrm{Calculando \textbf{a} e \textbf{h}:}\\\\ \mathrm{A_L=n.a.h\ \to\ 225=3.a.h\ \to\ 225=3.\dfrac{h}{3}.h}\\ \mathrm{225=h^2\ \to\ h=\sqrt{225}\ \to\ h=15\ m}\\\\ \mathrm{a=\dfrac{h}{3}\ \to\ a=\dfrac{15}{3}\ \to\ a=5\ m}$

$\textrm{Calculando a \'area da base:}\\\\ \mathrm{A_B=A_{\bigtriangleup eq.}\ \to\ A_B=\dfrac{a^2\sqrt{3}}{4}}\\ \mathrm{A_B=\dfrac{5^2\sqrt{3}}{4}\ \to\ A_B=\dfrac{25\sqrt{3}}{4}\ m^2}\\\\ \textrm{Calculando o volume do prisma:}\\\\ \mathrm{V=A_B.h\ \to\ V=\dfrac{25\sqrt{3}}{4}.15}\\ \mathbf{V=\dfrac{375\sqrt{3}}{4}\ m^3}$