Suppose v is the height of the lighthouse. OH is the distance to the horizon from the top of the lighthouse. Using a geometry thereom, the diameter squared = (diameter+h)h. You could also come up with a formula by using the Pythagorean Theorem. In that case, (v+r)squared = r squared + OH squared. And then the radius or diameter of the earth could be substituted in. Another extension could be to find formulas for distance to the horizon for other other planets. Formulas that are then derived usually are simplified by the fact that the value of v is so small compared to r.
Those formulas can be found on-line. Using one of them, by my calculation, to have the horizon be 30 miles away you would have to be 605 feet up in the air. Pretty tall. The Eiffel Tower is 984 feet. But the tallest now are less than 300 feet. So the video, or I seem to be off. But maybe not. It's certainly possible. You don't get picked as a wonder of the world if you're not pretty impressive.
