Circumscribed - sphere

A cube with a volume of 4096 cm³ is described and inscribed by a sphere. Calculate how many times the volume of the circumscribed sphere is greater than the inscribed sphere.

Final Answer:

k =  5.1962

Step-by-step explanation:

$$\begin{gathered} V=4096{\text{cm}}^3 \\ V=a^3 \\ a=\sqrt[ \\ 3]{V}=\sqrt[ \\ 3]{4096}=16\text{cm} \\ d=\sqrt{3}\cdot a=\sqrt{3}\cdot 16=16\sqrt{3}\text{cm}\doteq 27.7128\text{cm} \\ {r}_1=a/2=16/2=8\text{cm} \\ {r}_2=d/2=27.7128/2=8\sqrt{3}\text{cm}\doteq 13.8564\text{cm} \\ {V}_1=\frac{4}{3}\cdot \pi \cdot r_1^3=\frac{4}{3}\cdot 3.1416\cdot 8^3\doteq 2144.6606{\text{cm}}^3 \\ {V}_2=\frac{4}{3}\cdot \pi \cdot r_2^3=\frac{4}{3}\cdot 3.1416\cdot 13.8564^3\doteq 11143.9833{\text{cm}}^3 \\ k=V_2/V_1=11143.9833/2144.6606=3\sqrt{3}=5.1962 \end{gathered}$$

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