Chapter 11.8, Problem 2BE

To determine

Find the length of the curved portion of the ground track , arcAB.

Answer to Problem 2BE

20,944 feet

Explanation of Solution

Given:

The radius of the Heading Alignment Cylinder was 20,000 ft. and a typical measure for $\angle AMB=120°$ .

Calculation:

The given situation is :

  

Since M is exterior point and $\overline{MA}$ and $\overline{MB}$ are tangents to the circle.So, by Outside point Theorem,

  $\begin{array}{l}m\angle AMB=\frac{1}{2}\left(\text{far arc - near arc}\right)\\ \Rightarrow 120°= \frac{1}{2}\left(\left(360°-arcAB\right)-arcAB\right)\\ \Rightarrow 240°=360°-2arcAB\mathrm{.........................}[\text{multiply each side by 2]}\\ \Rightarrow 2arcAB=360°-240°\\ \Rightarrow arcAB=\frac{120°}{2}\\ \Rightarrow arcAB=60°\\ \Rightarrow m(arcAB)=\frac{60°}{360°}2\pi r\\ \Rightarrow m(arcAB)=\frac{1}{6}2\pi (20,000)\\ \Rightarrow \Rightarrow m(arcAB)=\frac{1}{3}\pi (20,000)=20,944\end{array}$

So, the curved portion of the ground track is 20,944 feet long.