Question

A area total de cubo é 288m2 calcule a diagonal e o volume?

Answer 1

Para o cubo, temos:

$\rightarrow~Area~do~Cubo~=~6\times Aresta^2\\\\\rightarrow~Diagonal~do~Cubo~=~Aresta\times\sqrt{3}\\\\\rightarrow~Volume~do~Cubo~=~Aresta^3$

Utilizando a formulação da área do cubo:

$Area~do~Cubo~=~6\times Aresta^2\\\\\\288~=~6\times Aresta^2\\\\\\Aresta^2~=~\frac{288}{6}\\\\\\Aresta~=~\sqrt{48}\\\\\\Aresta~=~\sqrt{2~.~2~.~2~.~2~.~3}\\\\\\Aresta~=~\sqrt{2^2~.~2^2~.~3}\\\\\\Aresta~=~2~.~2\sqrt{3}\\\\\\\boxed{Aresta~=~4\sqrt{3}~\,m}$

Agora, utilizando a formulação para a diagonal:

$Diagonal~=~Aresta\times\sqrt{3}\\\\\\Diagonal~=~4\sqrt{3}\times \sqrt{3}\\\\\\Diagonal~=~4~.~\sqrt{3}^2\\\\\\Diagonal~=~4~.~3\\\\\\\boxed{Diagonal~=~12~\,m}$

Por fim, utilizando a equação do volume:

$Volume~=~Aresta^3\\\\\\Volume~=~(4\sqrt{3})^3\\\\\\Volume~=~4^3~.~\sqrt{3}^{\,3}\\\\\\Volume~=~64~.~\sqrt{3}^{\,2}~.~\sqrt{3}\\\\\\Volume~=~64~.~3~.~\sqrt{3}\\\\\\\boxed{Volume~=~192\sqrt{3}~\,m^3}$