Distances on a circular ring road (Treegville)

Text

Treegville’s ring road is a circular loop of radius $5$ km centered at the origin $O$; Dustin Drive is the positive $x$-axis (east), Cupert Road is the positive $y$-axis (north). Jack lives where the ring road meets Morton Street, which leaves $O$ in the direction $\dfrac56\pi$ measured from Dustin Drive. Janet lives at the point $(3;-4)$ on the ring road; Astrid Avenue is the straight line joining her house to the center $O$.

(a) Find the angle between Astrid Avenue and Morton Street.

(b) Find the length of the path from Janet’s house to Jack’s house following the ring road.

(c) Find the length of the same path as the crow flies.

(d) Find the length of the path following streets parallel to Dustin Drive or Cupert Road.

Solution

Jack sits on the ring at angle $\dfrac56\pi=150°$: $J=(5\cos150°,\,5\sin150°)\approx(-4.33;\,2.5)$. Janet is at $(3;-4)$, which indeed lies on the circle since $3^2+4^2=25$; the angle of her radius is $\theta_J=\operatorname{atan2}(-4,3)\approx-53.13°$.

(a) The angle between rays $O$–Janet ($-53.13°$) and $O$–Jack ($150°$) is $150°-(-53.13°)=203.13°$; the convex angle (the smaller of the two) is $360°-203.13°\approx156.87°$.

(b) The central angle between the two points is $\approx156.87°=2.738$ rad; the arc length is $\ell=r\,\theta=5\cdot2.738\approx13.69$ km.

(c) The chord between the two points measures $\sqrt{(3-(-4.33))^2+(-4-2.5)^2}=\sqrt{95.98}\approx9.80$ km.

(d) The “grid” path (along streets parallel to the axes) measures $|\Delta x|+|\Delta y|\approx7.33+6.5\approx13.83$ km.

$\boxed{\text{angle}\approx156.87°;\ \text{arc}\approx13.69\text{ km};\ \text{chord}\approx9.80\text{ km};\ \text{grid}\approx13.83\text{ km}}$