Cubes

A sphere is inscribed in one cube and the same sphere is circumscribed about another cube. Calculate the difference between the volumes of the two cubes if the difference between their surface areas is 231 cm².

Final Answer:

x =  489.8 cm³

Step-by-step explanation:

$$\begin{gathered} S=231{\text{cm}}^2 \\ S=S_1-S_2 \\ {S}_1=4\pi r_1^2 \\ {S}_2=4\pi r_2^2 \\ 2r_1=\sqrt{3}a \\ 2r_2=a \\ S=3\pi a^2-\pi a^2=2\pi a^2 \\ a=\sqrt{\frac{S}{2\pi }}=\sqrt{\frac{231}{2\cdot 3.1416}}\doteq 6.0634\text{cm} \\ {r}_2=a/2=6.0634/2\doteq 3.0317\text{cm} \\ {r}_1=\sqrt{3}\cdot r_2=\sqrt{3}\cdot 3.0317\doteq 5.2511\text{cm} \\ {V}_1=\frac{4}{3}\cdot \pi \cdot r_1^3=\frac{4}{3}\cdot 3.1416\cdot 5.2511^3\doteq 606.497{\text{cm}}^3 \\ {V}_2=\frac{4}{3}\cdot \pi \cdot r_2^3=\frac{4}{3}\cdot 3.1416\cdot 3.0317^3\doteq 116.7204{\text{cm}}^3 \\ x=V_1-V_2=606.497-116.7204\doteq 489.7766{\text{cm}}^3 \\ \text{ Verifying Solution: } \\ {S}_1=4\pi \cdot r_1^2=4\cdot 3.1416\cdot 5.2511^2=\frac{693}{2}=346.5{\text{cm}}^2 \\ {S}_2=4\pi \cdot r_2^2=4\cdot 3.1416\cdot 3.0317^2=\frac{231}{2}=115.5{\text{cm}}^2 \\ \Delta S=S_1-S_2=\frac{693}{2}-\frac{231}{2}=\frac{693-231}{2}=\frac{462}{2}=231{\text{cm}}^2 \end{gathered}$$

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