#2             70. THE DOG FENCE               

The little moon Australis was as smooth as a billiard ball, making the landscape rather monotonous. However, the climate was pleasant, and it was decided to farm the arable lands to the south.

As the early farmers were constantly harrassed by fierce dogs from the north, the authorities agreed to erect a high fence along the whole length of the equator. This was a major undertaking, as the moon was 1000 km in diameter. The engineers were meticulous in their measurements, partitcularly for the supporting cable which would be strung at a constant height, two metres above the ground.

But when the cable arrived, it was discovered to be 12 metres shorter than expected.

“never mind,” said the chief engineer. “Twelve metres is a minute amount out of a total length of over 3,000,000 metres! It won’t matter if the fence is slightly under two metres high.”

Guess the height of the final fence, and then check your answer (by calculation!).
Hints and strategies

Hint 1      Hint 2       Solution 1       Solution 2       Extensions
HINT 1

We are interested in the length of the circular cable and its radius. How are these two quantities related?
HINT 2

Calculate/list the length and radius of the correct cable, the shorter cable, and the equator. What do you find?
SOLUTION

Surprisingly the difference between the the lengths of the planned cable and the equator is just over 12 metres. The erected fence had a height of only 0.16 metres – good for very small dogs!

A circle of radius R has circumference (length) 2R, so we obtain this table (in metres). Start from the grey entries and follow the arrows.

 
Radius
Circumference
Equator
500 000 >
3 141 592
Erected Fence
500 000.16
< 3 141 593
Planned Fence
500 002 >
3 141 605 ^

However, this answer is not exact. See Solution 2.
Solution 2

A better solution involves the use of some algebra. Let R be the radius of the planned fence, R – h the heght of the actual fence. Then calculating circumferences:

       2R – 2(R – h) = 12,
so       h = 12/2 = 1.91 metres

Hence the height of the new fence is 2 – 1.91 = 0.09 metres, suitable for only very small dogs!

Notice that if the cable had been 13 metres short, the construction would have been impossible!

Our two answers (0.16 and 0.09) are close, but different. Why is this so?
EXTENSION

1. The photo is of the moon Io. Where is this? How big is it? Could you grow crops on it?

2. One would have to suspect that the 0.09 answer is correct. Whenever you use a calculator, especially with numbers involving many digits, you always get rounding errors. This can give major problems with so-called iterative processes, where a calculated answer is the input of the next stage of the process. This gives problems with complicated problems like weather forecasting. You might be interested in looking up ‘chaos theory’ on the net.