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Puzzler from Steve:

Imagine a steel band snug to the Earth that encircles it entirely along the equator (about 25,000 miles around). You add 10 feet to the band to create a gap between the band and the Earth so that it is no longer snug. The gap is large enough:

a) For a person to crawl under
b) For a mouse to scamper under
c) To slip a pioece of paper under
d) For a baterium to swim under

The Answer:

To find the answer, we need to find the radii for two differently-sized circles. The smaller circle has a circumference of 25,000 miles, and the larger circle has a circumference of 25,000 + 10/5280 (25,000.0018939) miles.

We'll round off this larger circumference to 25,000.002 miles. Let's now find the radii for the two circles with the following formula:

Radius = Circumference/2 x pi

For our purposes, 3.14 is a good enough approximation for pi. So here goes:

Radius of Smaller Circle:

1) Radius = 25,000/2 x 3.14
2) Radius = 25,000/6.28
3) Radius = 3980.8917 miles, radius of the smaller circle

Radius of Larger Circle

1) Radius = 25,000.002/2 x 3.14
2) Radius = 25,000.002/6.28
3) Radius = 3980.8920 miles, the radius of the larger circle.

The Answer!

Subtract the smaller radius from the larger radius:

3980.8920 - 3980.8917 = 0.0003 miles. Because 0.0003 x 5,280 feet = 1.584 feet, the answer is a) large enough gap for a person to crawl under.

A Simpler Way!

My friend Roxana let me know of a much, much simpler way to solve the puzzle: Use this formula:

R (increase of radius) = C (increase of circumference)/2 x pi
R = C/2pi
R = 10/2 x 3.14
R = 10/6.28
R = 1.59 feet

Finis

Deriving the Formula for the Area of a Circle