Chapter 9.7, Problem 40ES

(a) The accompanying figure shows a sector of radius $r$ and central angle $2\alpha .$ Assuming that the angle $\alpha$ is small, use the local quadratic approximation of $\cos \alpha$ at $\alpha =0$ to show that $x\approx r{\alpha }^2/2.$

(b) Assuming that the Earth is a sphere of radius 4000 mi, use the result in part(a) to approximate the maximum amount by which a 100 mi arc along the equator will diverge from its chord.