## Sunday, July 27, 2008

### Men on the Moon

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.

First Question: How deep should this groove be to overcome the problem of the lacking 1 meter of cable?

A Hint : Assume that the moon's diameter is 3476000 meters.

Second Question: The agency's director considers digging the groove 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?

First Answer: The moon's diameter is not needed to solve this problem! Let the moon's radius be r meters. Then the length of the cable is 2×pi×r-1 meters. Let the depth of the groove be x meters. 2×pi×(r-x) must be equal to 2×pi×r-1. Solving the equation 2×pi×(r-x)=2×pi×r-1 gives x = 1/(2×pi) meters (about 0.159 meters).

Second Answer: From the first part of this puzzle, we know that the radius of the circle with a circumference of the cable's length is 1/(2×pi) meters less than the moon's radius. In the figure shown below, therefore

x = r - 1/(2×pi)

and

cos(a) = x / r = (r - 1/(2×pi)) / r

and, when taking a in radians,

y = (a / (2×pi)) × (2×pi×r) = a × r. Since r=3476000/2=1738000 meters, we can calculate that y is approximately 744 meters, which is the distance that the cable should be laid north of the moon's equator to settle the problem of the lacking 1 meter of cable.