Arch ground length

The arch has a radius of 3.3 m, a span of 3.25 m, and a height of 20 cm above the ground. What is the length of the arc to reach the ground?

Final Answer:

x =  4.1387 m

Step-by-step explanation:

$$\begin{gathered} r=3.3\text{m} \\ z=3.25\text{m} \\ h=20\text{cm}\to \text{m}=20:100\text{m}=0.2\text{m} \\ {h}_2=\sqrt{r^2-(z/2{)}^2}=\sqrt{3.3^2-(3.25/2{)}^2}\doteq 2.8722\text{m} \\ {h}_3=h_2-h=2.8722-0.2\doteq 2.6722\text{m} \\ {h}_3=r\cdot \sin \alpha \\ \alpha =\arcsin (h_3/r)=\arcsin (2.6722/3.3)\doteq 0.9437\text{rad} \\ \theta =\pi -2\cdot \alpha =3.1416-2\cdot 0.9437\doteq 1.2541\text{rad} \\ x=\theta \cdot r=1.2541\cdot 3.3=4.1387\text{m} \end{gathered}$$

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