Track arc

Two straight tracks are at an angle 126°. They are joined with a circular arc with a radius r=1110 m. How long will the arc be connecting these lines (L)? How far is the arc's center point from track crossings (x)?

Final Answer:

L =  1046.2 m
x =  1245.8 m

Step-by-step explanation:

$$\begin{gathered} \alpha =126{}^{\circ } \\ r=1110\text{m} \\ \beta =180-\alpha =180-126=54{}^{\circ } \\ B=\beta =54°=0.94248=3\pi /10 \\ L=r\cdot B=1110\cdot 0.9425=1046.2\text{m} \end{gathered}$$

$$\begin{gathered} \gamma =\alpha /2=126/2=63{}^{\circ } \\ x=\frac{r}{\sin \gamma }=\frac{r}{\sin 63°}=\frac{1110}{\sin 63°}=\frac{1110}{0.891007}=1245.8=1245.8\text{m} \end{gathered}$$

Try another example