Question

Qual a razão entre os volumes de uma semiesfera e um semicilindro em relação ao volume de um semicone, conforme a figura a seguir?

Answer 1

$\mathbf{Volume\ da\ semiesfera\ (V_1):}\\\\ \mathrm{V_1=\dfrac{1}{2}.V_{esf.}=\dfrac{1}{2}.\dfrac{4\pi r^3}{3}=\dfrac{2\pi r^3}{3}}\\\\ \mathbf{Volume\ do\ semicilindro\ (V_2):}\\\\ \mathrm{V_2=V_{cil.}=\pi r^2h=\pi r^2.r=\pi r^3}\\\\\ \mathbf{Volume\ do\ semicone\ (V_3):}\\\\ \mathrm{V_3=V_{cone}=\dfrac{\pi r^2h}{3}=\dfrac{\pi r^2.r}{3}=\dfrac{\pi r^3}{3}}$

$\mathbf{Raz\~ao\ entre\ V_1\ e\ V_3:}\\\\ \mathrm{\dfrac{V_1}{V_3}=\dfrac{\frac{2\pi r^3}{3}}{\frac{\pi r^3}{3}}=\dfrac{2\pi r^3}{3}.\dfrac{3}{\pi r^3}=2\ \to\ \boxed{\mathbf{\dfrac{V_1}{V_3}=2}}}\\\\ \mathbf{Raz\~ao\ entre\ V_2\ e\ V_3:}\\\\ \mathrm{\dfrac{V_2}{V_3}=\dfrac{\pi r^3}{\frac{\pi r^3}{3}}=\pi r^3.\dfrac{3}{\pi r^3}=3\ \to\ \boxed{\mathbf{\dfrac{V_2}{V_3}=3}}}$