# Enigmatic Code

Programming Enigma Puzzles

## Enigma 1318: Extended horizons

From New Scientist #2476, 4th December 2004

When Hurr was on her home planet Rizon she loved to stand at the edge of the sea and view the distant horizon. Her inbuilt range-calculator enabled her to work out how far she was from the horizon.

Now that Hurr has secretly landed on Earth and has mingled with the local population, she still slips down to the sea’s edge and views the horizon. But it turns out that on Earth the horizon is twice as far away from her as it was on Rizon.

The Earth’s radius is approximately 3960 miles. What, to the nearest hundred miles, is the radius of Rizon?

[enigma1318]

### One response to “Enigma 1318: Extended horizons”

1. Jim Randell 9 June 2014 at 8:31 am

By Pythagoras the square of the distance to the horizon on a planet of radius r from an altitude of h is:

d² = (h + r)² – r² = h² + 2hr

And on Earth (radius = R) the horizon is twice as far away as it is on Rizon (radius = r). So the square of the distance is 4 times as much:

h² + 2hR = 4(h² + 2hr)
2hR = 3h² + 8hr
r = (2hR – 3h²)/8h = R/4 – 3h/8

Unless Hurr is hundreds of miles tall (which would make mingling with the local population difficult) the R/4 term will be hugely greater than the 3h/8 term.

We are told the Earth has a radius of 3960 miles, so the radius of Rizon is 990 miles (less ⅜ of Hurr’s eye height).

So, to the nearest 100 miles, Rizon has a radius of 1000 miles.

Solution: The radius of Rizon is approximately 1000 miles.

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