## Tuesday, June 21, 2011

### Moon Cable

A large space agency has decided to build a base on the moon. For this purpose, a cable must be laid around the moon's equator. When the cable is laid, it turns out to be 1 meter short. In a quickly arranged meeting, it is decided to investigate the possibility to lay the whole cable in a groove.

How deep does the groove need to be to make this work?

The agency's director considers digging a groove, no matter how deep, around the entire circumference is too expensive. He suggests to lay the whole cable just a bit north of the equator. How many meters north of the moon's equator should the cable be laid to settle the problem of the lacking 1 meter of cable?

1. Is it 7/44th of a metre.

2. The groove needs to be 0.16 m deep. Since it's to expensive to dig such a long groove the cable should be laid 42616 meters north or south of the equator.

3. The circumference of a circle is 2πr. To make the circumference a meter shorter, subtract one: 2πr - 1. To get the circumference formula again, subtract 1 in the form of 2π/2π and factor out a 2π to get 2π(r - 1/2π). The trench has to be 1/2π m (about 0.16 m or half a foot) deep.

To find out how far north to move, construct a right triangle with hypotenuse r, long leg r - 1/2π, and short leg y, the amount north to move. Appeal to Pythagoras' theorem to find that y is √(r/π - (1/2π)²). The moon's radius is about 1700 km, a third of that is about 600, and the square root of that is about 24 m, about 79 feet.

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