Question

a space shuttle is 50 miles above earth an astronaut on the shuttle sees the sun rise over earth. to the nearest mile, what is the distance from the astronaut to the horizon? (hint: earths radius is about 4,000 miles)

Answer 1

From the astronaut to the horizon is a tangent line to the curvature of the earth.  And from that point on the horizon to the center of the earth is at a right angle to the tangent and is equal to the radius of the earth...so we can say

cosα=r/(r+h) where r is the radius of the earth and h is the height above the surface of the earth...

α=arccos(r/(r+h))

now tanα=d/r where d is the distance from the astronaut to the point on the horizon so:

tan(arccos(r/(r+h))=d/r

d=rtan(arccos(r/(r+h)))  and using r≈4000 and h=50 we get:

d=4000tan(arccos(4000/(4050)))

d≈634.42mi

d≈634 mi (to nearest mile)