Question

Assume the earth is a sphere

with radius 3960 miles, how high does the satellite have to be above the earth’s surface to see both Los
Angeles and Miami, 2700 miles apart

Answer 1

The length of the arc between L A and Miami = 2700 miles

length of arc = r*c where c is in radians

2700 = 3960 * c

c = 2700/3960 = 0.68182 radians+

The tangents from the satellite to the 2 points on the earth make a 90 degree angle with the radii of the earth, so we have 2 right angled triangles with one angle = 1/2 * 0.68182 = 0.34091 radians and the hypotenuse is the distance from center of the earth to the satellite.

if x is the distance of the satellite from the surface of the earth we have the equation

cos 0.34091 = 3960 /( x + 3960)

cos 0.34091(x + 3960) = 3960

x = (3960 - cos0.3409* 3960) / cos0.34091

= 241.8 miles Answer