Sketch this, assuming the Earth is a sphere with radius R, so a plane slice through the periscope, ship and center of the Earth is a circle of radius R. Draw the lines out from the center (O) of the circle to point A at the top of the periscope and point B at the top of the ship. so that line AB is tangent to the circle at point C. That makes triangles OAC and OBC right triangles, each having the right angle at C.
From the problem, the lengths OA = R+5, OC = R, and OB=R+50. Label the lengths AC = p and BC= q, then use Pythagoras:
R² + p² = (R + 5)²
R² + q² = (R + 50)²
Solve those:
p² = (R + 5)² - R² = 10R - 25
p = √(10R + 25)
q² = (R + 50)² - R² = 100R + 2500
q = √(100R + 2500)
Find a good value for the radius R (in ft. units!) and calculate. The distance from periscope top to ship top is (p + q) feet. Convert that to miles for your answer.
