Question

Determine o volume de um cubo inscrito em uma esfera cujo volume mede 2,304π cm³.

a) 1 536√3 cm³
b) 2 660√3 cm³
c)1,536√3 cm³
d) 15,36√3 cm³
e) 2,660√3 cm³ ​

Answer 1

$\largeO ~volume do ~cubo ~ \Rightarrow ~ V = 1536\sqrt{3} ~cm^3\\\\\\\largeLetra ~a) ~ 1536\sqrt{3} ~cm^3$

                          $\large Volume ~de ~s\acute{o}lidos ~geom\acute{e}tricos$

Encontrar o valor do raio da esfera.

$V = \dfrac{4}{3}~. ~\pi ~. ~r^3\\\\\\2304\pi = \dfrac{4}{3}~. ~\pi ~. ~r^3\\\\\\2304\pi ~. ~3 = 4 ~. ~r^3\\\\\\\dfrac{6912\pi }{\pi}=4 ~. ~r^3\\\\\\r^3 =\dfrac{6912}{4}\\\\\\r^3 = 1728\\\\r = \sqrt[3]{1728} \\\\r = 12 ~cm$

A diagonal do cubo é igual ao diâmetro da esfera:

$D = a\sqrt{3}$

Então o raio é igual:

$r =\dfrac{ a\sqrt{3} }{2}$

---

$r =\dfrac{ a\sqrt{3} }{2}\\\\\\12 = \dfrac{ a\sqrt{3} }{2}\\\\\\12 ~. ~2 = a\sqrt{3} \\\\\\a\sqrt{3} = 24\\\\a = \dfrac{24}{\sqrt{3} } \\\\\\a = 8\sqrt{3} ~cm$

Volume do cubo:

$V = a^3\\\\V = (8\sqrt{3} )^3\\\\V = 8^3 ~. ~(\sqrt{3} )^3\\\\V = 512 ~. ~3\sqrt{3} \\\\V = 1536\sqrt{3} ~cm^3$

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