Answer
The required solution is $2.5\ \text{radians}$ or $143.23{}^\circ $
Work Step by Step
The radius of the earth $ r $ is $4000\ \text{miles}$. The distance $ s $ between A and B is $10,000\ \text{miles}$.
The angle $\theta $ at the center between A and B is given by
$\theta =\frac{s}{r}$
Put $10,000\ \text{miles}$ for $ s $ and $4000\ \text{miles}$ for $ r $:
$\begin{align}
& \theta =\frac{10,000\ \text{miles}}{4000\ \text{miles}} \\
& =2.5\ \text{radians}
\end{align}$
Convert $\theta $ into degrees:
$\begin{align}
& \theta =2.5\ \text{radians}\left( \frac{180{}^\circ }{\pi \ \text{radians}} \right) \\
& =\left( \frac{2.5}{\pi } \right)180{}^\circ
\end{align}$
Put $\pi =3.14159$:
$\begin{align}
& \theta =\left( \frac{2.5}{3.14159} \right)180{}^\circ \\
& =\left( 0.79577 \right)180{}^\circ \\
& =143.23{}^\circ
\end{align}$
