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I See No Ships (Posted on 2005-01-23) Difficulty: 3 of 5
You are standing on a beach looking out into the ocean. Assuming your view is unobstructed, how far can you see?

No Solution Yet Submitted by Juggler    
Rating: 2.7500 (4 votes)

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Solution Three solutions | Comment 4 of 15 |

Answer #1:

Assume your eyes are 6 feet above the surface of the water.  Depending on your height, this allows more or less elevation of the land above the water's surface. Also assume the earth's radius is 3959 miles.

The horizon is then 3959*arccos(3959/(3959+6/5280)) = 2.99962... (about 3 miles) away--the farthest you can see of the surface of the ocean.

Added: Note that this is the distance along the curved surface of the earth, as is the next, rather than straight eyeball to distant object distance.

Answer #2:

Columbus is reputed to have argued that the earth can be demonstrated to be round by the fact that ships disappearing beyond the horizon become hidden by the horizon from bottom to top, so the last thing you see of them is the tops of their masts.  Depending on the height of the mast, the distance from its pinnacle to its own horizon would vary.  That distance would be added to the horizon distance of the observer, to get the total distance of the top of the mast to the observer.  The following table shows various mast heights in feet, the distance to the horizon from that point, and the total of that distance plus the remaining distance to the observer 6 feet above the water's surface:

 2   1.7318   4.7315
 4   2.4492   5.4488
 6   2.9996   5.9992
 8   3.4637   6.4633
10   3.8725   6.8721
12   4.2421   7.2417
14   4.5820   7.5816
16   4.8984   7.8980
18   5.1955   8.1951
20   5.4765   8.4762
22   5.7438   8.7435
24   5.9992   8.9989
26   6.2442   9.2438
28   6.4799   9.4795
30   6.7074   9.7070
32   6.9273   9.9269
34   7.1405  10.1401
36   7.3475  10.3472
38   7.5489  10.5485
40   7.7450  10.7446
42   7.9362  10.9359

Again, this is the foot-to-base distance along the surface of the earth.

Answer #3:

If you are looking at a sunrise or sunset, when the rays of the sun are sufficiently attenuated so you do not harm your eyes, you are looking at the sun at a distance of 93 million miles.

In fact, if it is nighttime and you see some stars,they are many light years away.  That's trillions of miles (using American trillions--10^12).  If you happen to be at a location where the Andromeda galaxy is visible near the horizon, that's thousands of times farther still.

Edited on January 23, 2005, 4:46 pm
  Posted by Charlie on 2005-01-23 16:32:06

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