The Lighthouse at Alexandria

I was watching a video about the Lighthouse at Alexandria. It no longer exists, but was one of the seven wonders of the world. The video said it could be seen from thirty miles away. They didn't mention how tall it was. That can be figured out, though. There is a formula. To the right is a diagram of the situation.

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.

Going for Two

Seems like I wrote something like this before, but things have changed now, so it is time to revisit. In the NFL, after you score a touchdown, you can go for a one-point conversion, by kicking, or go for a two-point conversion, by running or passing. The two pointers are harder to make, though. So which should you do? The kicks pretty much always went through. Because of that, the NFL decided they needed to make things a little more interesting and moved the kick back a ways. It still almost always goes through, but the percent is down a little bit - to a success rate of 93.8% of the time. The pass or throw option is roughly half - a success rate of 50.1%.

We should find the expected value for each. 

For a kick, you can get one point, 93.8% of the time:

                    1 x .938 = .938 points per try

For running or passing, you get two points, 47.9% of the time:

                    2 x .501 = 1.02 points per try

Is that even worth messing with? It depends on how good your team is, but typically a team scores something around 60 touchdowns in a season. That would mean roughly (.938 x 60 =) 56.28 points if you kick all the time. And (1.02 x 60 =) 61.2 when running or passing. So about five point over the course of a season. So not a lot, but then again, it only takes one point to lose a game.

And of course your decision could depend on the situation in any particular game. If you score a touchdown near the end of the game and you are now behind by two, you definitely would go for the two-point attempt. 

By the way, I don't know if this was in the book, but Scorecasting is a great little book about sports and how coaches don't always do what makes the most sense.