Question 292714
The shuttle, the point on the horizon and the center of the earth create a right triangle with right angle where the radius intersects the tangent line from the shuttle to the horizon. The radius is 4000 and the hypotenuse from the center of the earth to the shuttle is 4000+180. Let d be the distance from the shuttle to the horizon. By the Pythagorean Theorem we have then:

4180^2 = 4000^2 + d^2 
d^2 = 4180^2 - 4000^2

Solve for d.