Spherical cap

From the sphere of radius 11 was truncated spherical cap. Its height is 6. What part of the volume is a spherical cap from the whole sphere?

Correct answer:

x =  18.2569 %

Step-by-step explanation:

$$\begin{gathered} r=11\text{ cm } \\ v=6\text{ cm } \\ {\rho }^2=r^2-(r-v{)}^2 \\ \rho =\sqrt{r^2-(r-v{)}^2}=\sqrt{11^2-(11-6{)}^2}=4\sqrt{6}\text{ cm}\doteq 9.798\text{ cm } \\ {V}_1={\displaystyle \frac{\pi }{6}}\cdot v\cdot (3\cdot {\rho }^2+v^2)={\displaystyle \frac{3.1416}{6}}\cdot 6\cdot (3\cdot 9.798^2+6^2)\doteq 1017.876{\text{cm}}^3 \\ {V}_2={\displaystyle \frac{4}{3}}\cdot \pi \cdot r^3={\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 11^3\doteq 5575.2798{\text{cm}}^3 \\ x=100\cdot {\displaystyle \frac{V_1}{V_2}}=100\cdot {\displaystyle \frac{1017.876}{5575.2798}}=18.2569\mathrm{\%} \end{gathered}$$

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