Two hemispheres

In a wooden hemisphere with a radius r = 1, a hemispherical depression with a radius r/2 was created so that the bases of both hemispheres lie in the same plane. What is the surface of the created body (including the surface of the depression)?

Correct answer:

S =  10.2102 cm²

Step-by-step explanation:

$$\begin{gathered} r=1 \\ {r}_2=r\mathrm{/}2=1\mathrm{/}2={\displaystyle \frac{1}{2}}=0.5 \\ {S}_1=\pi \cdot r^2=3.1416\cdot 1^2\doteq 3.1416 \\ {S}_2=\pi \cdot r_2^2=3.1416\cdot 0.5^2\doteq 0.7854 \\ {S}_{11}=4\pi \cdot r^2\mathrm{/}2=4\cdot 3.1416\cdot 1^2\mathrm{/}2\doteq 6.2832 \\ {S}_{22}=4\pi \cdot r_2^2\mathrm{/}2=4\cdot 3.1416\cdot 0.5^2\mathrm{/}2\doteq 1.5708 \\ S=S_{11}+S_{22}+S_1-S_2=6.2832+1.5708+3.1416-0.7854\doteq 10.2102 \\ \text{ Verifying Solution: } \\ q=S\mathrm{/}\pi \mathrm{/}{r}^2=10.2102\mathrm{/}3.1416\mathrm{/}{1}^2={\displaystyle \frac{13}{4}}=3.25 \\ S={\displaystyle \frac{13}{4}}\cdot \pi \cdot r^2=10.2102{\text{cm}}^2 \end{gathered}$$

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