Problem

A pilot flying at 3 miles above the ground at 600 miles per hour sights the airport with a spotting scope. How fast must she turn the scope when the angle between the path and plane is 40^o to keep the scope pointed at the airport?

Solution

The solution is to find an expression for dq/dt, the change in the angle between the plane's path and the line of sight to the airport.

The pilot's velocity is

The distance to the airport, x, satisfies the equation

so

Take the derivative to get

Set these two equations for dx/dt equal to each other, and solve for dq/dt. We include the units so we can convert the answer to radians and seconds.