Question

Determine o volume de um cubo inscrito em uma esfera cujo volume mede 2,304πcm³

Answer 1

O volume de uma esfera de raio (R) é determinado pela formula:

$\mathsf{V_{esfera}=\dfrac{4}{3}\pi R^3}$

= = = = =

Foi informado que o volume da esfera é $\mathsf{2,304\pi \:cm^3}$. Logo:

$\mathsf{V_{esfera}=2,304\pi \:cm^3}\\\\\\\mathsf{2,304\diagup\!\!\!\!\!\pi =\dfrac{4}{3}\diagup\!\!\!\!\!\pi R^3}\\\\\\\mathsf{2,304=\dfrac{4R^3}{3}}\\\\\\\boxed{\mathsf{R^3=\dfrac{2,304\cdot 3}{4}}}\:\:\:\:\:\:\:\:\mathsf{(i)}$

= = = = =

Aplicando o teorema de Pitágoras na Figura 2 determinaremos que o valor da diagonal da base do quadrado:

$\mathsf{d^2=l^2+l^2}\\\\\mathsf{d^2=2l^2}\\\\\mathsf{d=\sqrt{2l^2}}\\\\\mathsf{d=l\sqrt{2}}$

= = = = =

A diagonal (D) do cubo equivale ao valor da diagonal (2R) da esfera. Aplicando o teorema de Pitágoras na Figura 1:

$\mathsf{D^2=l^2+d^2}\\\\\mathsf{\left(2R\right)^2=l^2+\left(l\sqrt{2}\right)^2}\\\\\mathsf{4R^2=l^2+2l^2}\\\\\mathsf{4R^2=3l^2}\\\\\mathsf{R^2=\dfrac{3l^2}{4}}\\\\\\\mathsf{R=\sqrt{\dfrac{3l^2}{4}}}\\\\\\\mathsf{R=\dfrac{l\sqrt{3}}{2}}$

Elevando os dois termos ao cubo teremos:

$\mathsf{R^3=\left(\dfrac{l\sqrt{3}}{2}\right)^3}\\\\\\\mathsf{R^3=\dfrac{l^3\left(\sqrt{3}\right)^3}{2^3}}\\\\\\\boxed{\mathsf{R^3=\dfrac{3l^3\sqrt{3}}{8}}}\:\:\:\:\:\:\:\:\mathsf{(ii)}$

= = = = =

O volume do cubo é determinado pela formula:

$\mathsf{V_{cubo}=l^3}$

= = = = =

Igualando i e ii:

$\mathsf{\dfrac{2,304\cdot 3}{4}=\dfrac{3l^3\sqrt{3}}{8}}\\\\\\\mathsf{\dfrac{2,304\cdot \diagup\!\!\!\!3}{\diagup\!\!\!\!4}=\dfrac{\diagup\!\!\!\!3\cdot l^3\sqrt{3}}{2\cdot \diagup\!\!\!\!4}}\\\\\\\mathsf{2,304=\dfrac{l^3\sqrt{3}}{2}}\\\\\\\mathsf{2,304\cdot 2=l^3\sqrt{3}}\\\\\\\mathsf{l^3=\dfrac{2,304\cdot 2}{\sqrt{3}}}\\\\\\\mathsf{l^3=\dfrac{4,608}{\sqrt{3}}}\\\\\\\boxed{\mathsf{V_{cubo}=\dfrac{4,608}{\sqrt{3}}}}$

= = = = =

Racionalizando teremos:

$\mathsf{V_{cubo}=\dfrac{4,608}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}}\\\\\\\mathsf{V_{cubo}=\dfrac{4,608\sqrt{3}}{3}}\\\\\\\boxed{\mathsf{V_{cubo}=1,536\sqrt{3}}}$

= = = = =

$\boxed{\boxed{\mathsf{Resposta:\:V_{cubo}=1,536\sqrt{3}\:\:\:ou\:\:\:\dfrac{4,608}{\sqrt{3}}}}}\: \: \checkmark$

Answer 2

Resposta:

192√3/125 cm³

Explicação passo-a-passo:

V=4/3πr³

2,304π=4/3πr³

r³=2,304π/4/3π

r³=2,304/4/3

r³=2304/1000÷4/3

r³=2304/1000*3/4

r³=6912/4000

r³=216/125

r=³√216/125

r=6/5

A diagonal do cubo será o diâmetro da esfera.

D=12/5

12/5²=l√2²+l²

144/25=2l²+l²

144/25=3l²

l²=144/25÷3

L=√144/75

L=12/5√3

L=4√3/5

(4√3/5)³

192√3/125