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# Treebeard's Stumper Answer9 Jan 98

Anacapa Mirage

On Christmas Day, my family drove down the coast to Santa Monica to visit my parents. It was a beautiful day, and we had a great view of the Channel Islands. Arch Rock is the isolated rocky islet on the east end of Anacapa Island, about eleven miles off the coast. It has a huge sea cave going right through it, like a giant keyhole. Between Rincon and Ventura, we noticed that we could clearly see the keyhole whenever the highway climbed the hillside above the ocean. But the cave disappeared whenever the road dropped back to sea level. Explain!

The sea cave on Anacapa Island is only visible when the highway climbs above the ocean because the Earth is curved! When the highway drops back to sea level, the cave disappears below the curve of the Earth, and we can only see the very top of the island. "Anacapa" is the native Chumash word for mirage, and perhaps this is what they had in mind. Of course, sailors can see further from the crow's nest atop the mast than from the deck. The curvature of the Earth must be obvious to any seafaring people, as it was to the ancient Greeks. The real stumper is why anyone would seriously think otherwise.

Graybear (aka Charles L. Smith III) adds another example:

Curvature of the Earth? If so, an East Coast version occurred when they built the Tacoma-Narrows Bridge in New York, which was at the time the longest suspension bridge by far. I heard a story that when they built the two towers, they couldn't understand why the tops were so much farther apart than the bottoms - after all, weren't the towers perfectly plumb? The curvature of the earth had not been considered. As man builds larger and larger items, everyday physical circumstances affect the outcome, but it often takes awhile to see the cause-and-effect relationship.

How far can we see to the horizon? We can find an useful approximation using just the Pythagorean Theorem.

If R is the radius of the Earth (about 4000 miles), and h is our height (in feet) above the surface, and d is the distance to the horizon that we want to find (in miles), then

```   d^2 + R^2 = (R+h)^2
d^2 + R^2 = R^2 + 2Rh + h^2
d^2 = 2Rh + h^2
d = sqrt{h(2R + h)}

But h is tiny compared with R, so we can approximate

d = sqrt{2Rh}

What about units?  It's handy to measure h in feet, and R and d in miles, so

h ft
d = sqrt{2  ------------  4000 miles}
5280 ft/mile

2 x 4000
d = sqrt{-------- h} miles
5280

d = sqrt{1.5 h} miles
or
d = 1.2 x sqrt{h} miles

```
So from an eye height of 5 feet, we can only see about 2.7 miles. But from a height of 85 feet, we can see ll miles, all the way to the base of Anacapa Island.

(It sure is a pain trying to show math in HTML!)

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