Circular railway

The railway is to interconnect in a circular arc the points A, B, and C, whose distances are | AB | = 30 km, AC = 95 km, BC | = 70 km. How long will the track from A to C?

Correct result:

b1 =  103.11 km

Solution:

$c=30 \ \\ b=95 \ \\ a=70 \ \\ \ \\ s=\dfrac{ a+b+c }{ 2 }=\dfrac{ 70+95+30 }{ 2 }=\dfrac{ 195 }{ 2 }=97.5 \ \\ S=\sqrt{ s \cdot \ (s-a) \cdot \ (s-b) \cdot \ (s-c) }=\sqrt{ 97.5 \cdot \ (97.5-70) \cdot \ (97.5-95) \cdot \ (97.5-30) } \doteq 672.6522 \ \text{km}^2 \ \\ \ \\ r=S/s=672.6522/97.5 \doteq 6.899 \ \text{km} \ \\ \ \\ R=\dfrac{ a \cdot \ b \cdot \ c }{ 4 \cdot \ r \cdot \ s }=\dfrac{ 70 \cdot \ 95 \cdot \ 30 }{ 4 \cdot \ 6.899 \cdot \ 97.5 } \doteq 74.1468 \ \text{km} \ \\ \ \\ \Delta R-R-b \ \\ b^2=R^2+R^2 - 2 \ R ^2 \ \cos θ \ \\ \ \\ θ=\arccos(\dfrac{ 2 \cdot \ R^2 - b^2 }{ 2 \cdot \ R^2 } )=\arccos(\dfrac{ 2 \cdot \ 74.1468^2 - 95^2 }{ 2 \cdot \ 74.1468^2 } ) \doteq 1.3906 \ \text{rad} \ \\ \ \\ b_{1}=θ \cdot \ R=1.3906 \cdot \ 74.1468=103.11 \ \text{km}$

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