Spherical cap

Place a part of the sphere on a 4.6 cm cylinder so that the surface of this section is 20 cm². Determine the radius r of the sphere from which the spherical cap was cut.

Correct result:

r =  3.0683 cm

Solution:

$$\begin{gathered} D=4.6\text{ cm } \\ R=D\mathrm{/}2=4.6\mathrm{/}2={\displaystyle \frac{23}{10}}=2.3\text{ cm } \\ S=20{\text{cm}}^2 \\ S=2\pi r\cdot v\text{ cm } \\ {r}^2=R^2+(r-v{)}^2 \\ {v}^2-2rv+R^2=0 \\ v={\displaystyle \frac{S}{2\pi \cdot r}} \\ r={\displaystyle \frac{S}{\sqrt{4\pi \cdot S-4{\pi }^2\cdot R^2}}}={\displaystyle \frac{20}{\sqrt{4\cdot 3.1416\cdot 20-4\cdot 3.1416^2\cdot 2.3^2}}}\doteq 3.0683=3.0683\text{ cm } \\ \text{ Verifying Solution: } \\ v=r-\sqrt{r^2-R^2}=3.0683-\sqrt{3.0683^2-2.3^2}\doteq 1.0374\text{ cm } \\ {S}_1=2\pi \cdot r\cdot v=2\cdot 3.1416\cdot 3.0683\cdot 1.0374=20{\text{cm}}^2 \\ {S}_1=S \end{gathered}$$

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