A periscope is 5 feet above the surface of the ocean Through

A periscope is 5 feet above the surface of the ocean. Through it can be seen a ship that rises to 50 feet above the water. To the nearest mile, the farthest away that ship could be is how many miles?

Solution

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.

R = Radius of earth = 21 x10^6 feet

p = 14491 feet

q = 45826 feet

p+q = 60317 feet = 11.42 miles