Viewing Distance to the Horizon
The ocean is a fun place to contemplate what you can see. How far can you see if you're standing on the beach? What if you're standing on the deck of a cruise ship?
First, the beach. The way to calculate this is to use a little bit of trigonometry. You knew that class would come in handy someday! As you remember, (heh heh), the Pythagorean theorem says that a triangle that has one right angle has a relationship between the three sides. A right angle is one that is 90 degrees. In this case, the triangle is between your eyeballs, the edge of the horizon, and the molten center of the planet Earth.
We know that the distance between the edge of the horizon and the center of the earth is 3,960 miles. That is the radius of the earth from every spot in the ocean :) Next, we know the distance between your eyeballs and the center of the earth - that's 3,960 miles plus your height. So let's say you're 6' for a nice round number. In miles, your eyeballs are .001136 above the beach's surface. That's 3,960.001136 total between your eyes and the earth's core.
Now, the Pythagorean theorem says that X2 + Y2 = Z2. X is the distance between you and the horizon. Y is the distance between the horizon and the earth's core, and Z is the distance between your eyeballs and the earth's core. You have formed a giant right triangle, with the right angle being between the horizon, the core, and your eyeballs.
So when you say X2 + 3,9602 = 3,960.0011362 it turns out that X is 2.9995 miles. I think we can round that to 3 miles. So from the edge of the water, you can see 3 miles out to the horizon.
When you're up on a ship's deck, you're much higher off the ocean surface. So your eyeball's height gets much higher. If we assume a 60' deck, and a 6' you, that means you can see about 9.5 miles in any given direction.
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