Path lengths on the ring road

Statement

The hospital (point $A$) lies on a ring road of radius $3$ km, at an angle of $70°$ with respect to the $x$-axis (centre at the origin). The police station (point $B$) lies on the same ring road, at an angle of $-30°$. Determine the length of the paths from $A$ to $B$:

• path 1: from $A$ to $B$ in a straight line;
• path 2: from $A$ to $B$ along the ring road;
• path 3: from $A$ to the centre, and from the centre to $B$;
• path 4: first along the $y$-axis, then along the $x$-axis.

Solution

Coordinates of the two points on the circle of radius $3$: $A=(3\cos 70°,\,3\sin 70°)\approx(1.026;\,2.819),\qquad B=(3\cos(-30°),\,3\sin(-30°))\approx(2.598;\,-1.5).$ The central angle between $A$ and $B$ is $\Delta\theta=70°-(-30°)=100°$.

Path 1 (chord). Using the chord formula $\ell=2R\sin\dfrac{\Delta\theta}{2}$: $p_1=2\cdot 3\cdot\sin 50°\approx 4.60\ \text{km}.$

Path 2 (arc). $p_2=R\,\Delta\theta_{\text{rad}}=3\cdot\dfrac{100\pi}{180}\approx 5.24\ \text{km}.$

Path 3 (through the centre). Two radii: $p_3=R+R=3+3=6\ \text{km}.$

Path 4 (along the axes). The “staircase” path: $p_4=|x_A-x_B|+|y_A-y_B|\approx 1.572+4.319\approx 5.89\ \text{km}.$

$\boxed{p_1\approx 4.60;\quad p_2\approx 5.24;\quad p_3=6;\quad p_4\approx 5.89\ \ (\text{km})}$