Chapter 2.5, Problem 34ES

An Earth-observing satellite can see only a portion of the Earth’s surface. The satellite has horizon sensors that can detect the angle $\theta$ shown in the accompanying figure. Let $r$ be the radius of the Earth (assumed spherical) and $h$ the distance of the satellite from the Earth’s surface.

(a) Show that $h=r\left(\csc \theta -1\right)$ .

(b) Using $r=6378\text{ km}$ , find the rate at which $h$ is changing with respect to $\theta$ when $\theta =30°$ . Express the answer in units of kilometers/degree.