Question

A space camera circles the Earth at a height of h miles above the surface. Suppose that d distance, IN MILES, on the surface of the Earth can be seen from the camera.

(a) Find an equation that relates the central angle theta to the height h.

(b) Find an equation that relates the observable distance d and theta.

Answer 1

Draw a circle with the camera a distance h above it. Draw lines tangent from the camera location (P) to the two sides of the Earth (with radius R). Also draw lines from the points of tangency to the center of the Earth. You should have two congruent right triangles, each with a hypotenuse h + R. Let theta be the angle subtended by observed portion of the Earth as seen from the center of the Earth. I don't know if that is what they call the "central angle" or not. It will be bisected by the line from the camera to the center of the Earth

(a) cos (theta/2) = R/(R +h)

(b) d = theta * R
where theta is in radians