Drilling Rig Visibility

The image shows an oil rig at sea which rises 500 feet above sea level and is located 50 "surface" miles from shore. Can the rig be seen from a restaurant atop an 800 foot hotel at the shoreline?

Prove: Depending upon weather, "the rig might be seen," or that it will not be seen.

To get at it, lets assume some important lady at dinner in the resturant sees the gas flare at the rig and screams, "EGad! Polution!" By geometry we calculate the distance then compare it to the stated 50 "surface" miles to sea.

Calculate: the ground-sea level distance from the rig to shore.
♦ The exaggerated sketch depicts the very limit of visibility of the rig (C) from the top floor of the hotel (A). The radius of Earth is represented as R.

The distance we seek is the length of the Earth-surface arc extending: E-B-D.

To proceed toward a solution, the arc lengths, EB and BD are determined then added. If this length is less than 50 miles, the rig can be seen. Otherwise the rig can not be seen.

The arc length, s_EB is determined from the arc length formula with the angle, uppercase theta, θ:

(1)This is the formula for the length of a circular arc.

A sketch of the the first triangle, 0AB shows:

(2) Definition of the cosine of θ.

Therefore,

(3)Using the inverse cosine function.

The first part of the arc length (s) [extending from shore (E) to horizon (B)] is determined as:

(4) Most calculators have an inverse
cosine operation.

Here it is wise to pause, to reflect. What do we sek to accomplish? Our sketch is enormously exaggerated to present the physical idea. The radius of Earth, R in feet being about 3963mil x 5280feet/mil, is enormous in comparison to the height of the hotel, 800 feet, or drill rig. So the theory of our solution is correct but most calculators will be unable to deal with this discrepancy in numbers.