Two hemispheres

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

Final Answer:

S =  10.2102 cm²

Step-by-step explanation:

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

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