A rope around the Earth
A rope is wrapped tightly around the Earth's equator (treat the Earth as a perfect sphere). By how many metres must the rope be lengthened so that it floats 1 metre above the ground all the way around?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
You do not need the Earth's radius. Call it .
Compare the circumferences of circles with radii and .
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The rope's length is the circumference radius, which is linear in the radius. Adding 1 metre to the radius adds metres to the length, whatever the radius is.
Step 1. Tight, the rope has length .
Step 2. One metre above the ground, it must have length .
Step 3. The difference is metres.
The surprise. The answer is the same for a football or for the Earth. The reverse is just as striking: adding 1 metre of rope to the Earth's 40,000 km lifts it by cm all the way round.