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Kentucky visible from Chattanooga? *July 22, 2007*

*Posted by eric22222 in General, Math.*

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[EDIT: fixed pictures]

[RE-EDIT: missed step in calculations]

Dad, Ashley, and I went to Chattanooga for the weekend. We saw the aquarium, the IMAX, and Rock City. We also did some mini-golf. It was a good trip, I’d say.

I started thinking today about the curvature of the Earth. At Rock City, they say you can see Kentucky from the overlook. I doubted this, so today I got to work on making a formula for what I’m looking for. Trig is about ten times easier with drawings, so I’ve made sure to include some diagrams. Here we go:

From any two points on the Earth, your view is blocked by the horizon. The planet sort of bulges between any two points, just because that’s how circles work. The higher your vantage point, the more you can see. For example, you can see more if you’re on a mountain. Now, if we have our two points already (in this case, the southern and northern borders of Tennessee), there will be a set height we need to see over. That’s the red line in the diagram above.

To start, we’ll just do a simple 90 degree difference. If we extend the distance of the red line to the circle’s center, we form a triangle. To get X, we use some simple trig: the cosine of 45 degrees will equal X over R, where R is the circle’s radius. Now we can see that our red line height is equal to the radius minus X. So now we see that the height equals R-R*cos(theta), or R(1-cos). Back to Rock City, then.

According to Wikipedia, Tennessee extends from 35°N to 36°41’N. That’s **1°41′** (if we didn’t have the degrees, we could use the distance divided by the Earth’s circumference times 360°). Half that (since we want the height at the midpoint) is **0°50′** Anyway, using average values for the Earth’s radius (3959.87 miles), we get **0.43 miles** (about 2270 feet). So the curvature of the Earth from the bottom of Tennessee to its top creates a wall almost half a mile high. Even though Rock City is on a mountain, it’s only 1500 feet above sea level. That’s about 1/4 of a mile.

So there is no possible way that you can see Kentucky from Rock City (same goes for Virginia).

Eric, you have absolutely too much time on your hands. Plus, you made my brain hurt. I’m going to go punch a rock until things make sense again.