Imagine you would have to put a rope around the moon. Since the moon is 1,738,000 metres in diameter, this is a hard task. Finally you have managed to get the rope around the moon but... it is one meter short.

You decide to dig a groove all around the moon, so that the shorter rope suffices. How deep must this groove be? (Assume the Moon to be a perfect sphere.)

Every circle fits the formula C = pi * D.

If you know D, you can find C.  If you know C you can find D.

If you want to find diffences between 2 circles, use the relationships, ie

C - C' = (pi * D) - (pi * D') = pi (D - D') = 2pi (R - R')

In this problem, we know C - C' = 1, so

1 = (pi * D) - (pi * D')
1 = pi (D - D')
1 = pi (2R - 2R')
1 = 2pi (R - R')
1/2pi = R - R'

R - R' is the depth of the groove, and it will always be equal to 1/2pi * (C - C').  In this case, (C - C') is 1, so the answer is 1/2pi. Had we come up 2 meters short, we would need to dig the groove 1/2pi * 2  = 1/pi meters.

If you have globe on your desk, and stretch a string around it and come up 1 inch short, you will need to make a grove 1/2pi inches to make it fit.

This is a good problem to show that you don't have to do a lot of brute force number crunching if you know the relationships.

Put a belt around the Earth. How much do you have to let it out so that it is 1 foot off the ground all around?

R - R' = 1 foot

C - C' = pi (D - D') = pi (2), so you have to let out the belt 2pi feet.

Posted by bob909 on 2004-10-27 09:44:21