Archaeologists

Archaeologists need to find out the size of the vessel if the sherd found was in the shape of a circular section with a length of 12 cm and a height of 3 cm. What is the area of this section?

Final Answer:

r =  7.5 cm
S =  25.1604 cm²

Step-by-step explanation:

$$\begin{gathered} l=12\text{cm} \\ h=3\text{cm} \\ {r}^2=(l/2{)}^2+(r-h{)}^2 \\ {r}^2=(l/2{)}^2+r^2-2hr+h^2 \\ 0=(l/2{)}^2-2hr+h^2 \\ 2hr=l^2/4+h^2 \\ r=\frac{l^2/4+h^2}{2\cdot h}=\frac{12^2/4+3^2}{2\cdot 3}=7.5\text{cm} \end{gathered}$$

$$\begin{gathered} \sin \alpha /2=l/2:r=12/2:7.5=\frac{4}{5}=0.8=4:5 \\ \alpha =2\cdot \arcsin (\frac{l}{2\cdot r})=2\cdot \arcsin (\frac{12}{2\cdot 7.5})\doteq 1.8546\text{rad} \\ \Phi =\alpha \to \text{°}=\alpha \cdot \frac{180}{\pi }\text{°}=1.8546\cdot \frac{180}{\pi }\text{°}=106.2602\text{°} \\ {S}_1=\frac{l}{2}\cdot (r-h)=\frac{12}{2}\cdot (7.5-3)=27{\text{cm}}^2 \\ {S}_2=\pi \cdot r^2\cdot \frac{\alpha }{2\pi }=3.1416\cdot 7.5^2\cdot \frac{1.8546}{2\cdot 3.1416}\doteq 52.1604 \\ S=S_2-S_1=52.1604-27\doteq 25.1604{\text{cm}}^2 \\ \text{ Verifying Solution: } \\ {r}_2=\sqrt{(l/2{)}^2+(r-h{)}^2}=\sqrt{(12/2{)}^2+(7.5-3{)}^2}=\frac{15}{2}=7.5\text{cm} \end{gathered}$$

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