Chapter 3.4, Problem 23ES

A satellite is in an elliptical orbit around the Earth. Its distance $r$ (in miles) from the center of the Earth is given by

$r=\frac{4995}{1+0.12cos\theta }$

where $\theta$ is the angle measured from the point on the orbit nearest the Earth’s surface (see the accompanying figure).

(a) Find the altitude of the satellite at perigee (the point nearest the surface of the Earth) and at apogee (the point farthest from the surface of the Earth). Use $3960\text{ mi}$ as the radius of the Earth.

(b) At the instant when $\theta$ is $120°$ , the angle $\theta$ is increasing at the rate of $2.7°\text{/min}$ . Find the altitude of the satellite and the rate at which the altitude is changing at this instant. Express the rate in units of mi/min.