Cube in sphere

The cube is inscribed in a sphere with a radius r = 6 cm. What percentage is the volume of the cube from the volume of the ball?

Correct answer:

p =  36.7553 %

Step-by-step explanation:

$$\begin{gathered} r=6\text{ cm } \\ D=2\cdot r=2\cdot 6=12\text{ cm } \\ u=D=12\text{ cm } \\ u=\sqrt{3}a \\ a=u\mathrm{/}\sqrt{3}=12\mathrm{/}\sqrt{3}=4\sqrt{3}\text{ cm}\doteq 6.9282\text{ cm } \\ {V}_1=a^3=6.9282^3=192\sqrt{3}{\text{cm}}^3\doteq 332.5538{\text{cm}}^3 \\ {V}_2={\displaystyle \frac{4}{3}}\cdot \pi \cdot r^3={\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 6^3\doteq 904.7787{\text{cm}}^3 \\ p=100\cdot {\displaystyle \frac{V_1}{V_2}}=100\cdot {\displaystyle \frac{332.5538}{904.7787}}=36.7553\mathrm{\%} \end{gathered}$$

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