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# From a height of six feet above sea level, looking out onto a clear sea, how far away is the horizon?

Question #81605. Asked by gmackematix. (Jun 06 07 8:32 PM)

mutchisman

According to this web-page the calculation is made by finding the square root of the height of your eyes (in feet) and multiplying that value by 1.17.

The square root of 6 is 2.45; multiply that by 1.17 and you get an answer of 2.86 nautical miles.

http://www.boatsafe.com/kids/distance.htm

 Jun 06 07, 11:31 PM
davejacobs

Or 3.29 real miles!

 Jun 07 07, 8:18 AM
gmackematix

The situation involves a right angled triangle involving the distances from the eyes to the centre of the Earth (R+h), from the horizon to the centre of the earth (R) and the required distance to the horizon x. Suppose all distances are in feet.
By Pythagoras, x = Sq rt ((R+h)^2 - R^2)
= Sq rt (2Rh + h^2)
Given that R is the radius of the Earth, then h^2 is so small compared to 2Rh ignoring it won't affect our approximate result.
So x ~ Sq rt (2Rh) = Sq rt (2R) * Sq rt h
The radius of the Earth is about 3,960 miles or 3,960 * 5,280 = 20,908,800 feet.
So Sq rt (2R) ~ 6,467.
So x in miles ~ 6,467/5,280 * Sq rt h ~ 1.22 * Sq rt h.
For h = 6 ft, I make x to be as near as dammit 3 miles.

I guess what I'm trying to say is yay, Mutch!

 Jun 07 07, 7:50 PM

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