Cubes

One cube is an inscribed sphere, and the other one is described. Calculate the difference of volumes of cubes if the difference of surfaces in 231 cm².

Correct 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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