Problem #87 EASY
The Extra Rope Around the Earth
Paradox Geometry Math Limits
Problem Statement
A rope is tied tightly around the equator of the Earth, lying flat on the surface. The Earth's radius is approximately 6,371 km. Now the rope is extended so that it floats exactly 1 metre above the entire equator — a perfectly uniform gap all the way around. How much extra rope is needed? Your first instinct will tell you it must be thousands of metres. Compute the exact answer and prepare to be astonished.
Answer & Quick Explanation
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Exactly 2π ≈ 6.283 metres of extra rope. The Earth's radius is irrelevant — the answer depends only on the increase in radius (1 metre). Formula: extra rope = 2π × ΔR, completely independent of R.
WOW Moment:
- The Earth has a circumference of ~40,074 km.
- You only need 6.28 extra metres of rope to lift the entire rope 1 full metre off the ground all the way around the planet.
- That is about the length of a tall man lying down.
- The same 6.28 metres would also raise a rope around a marble by exactly 1 metre. Or around the Sun. Or around the observable universe.
- The size of the circle is completely irrelevant. Extra rope = 2π ≈ 6.28 m, always, forever, for any circle.
Detailed Editorial Solution
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This is one of the most counterintuitive results in elementary mathematics. Our intuition says a rope around the Earth must be enormously long, so adding 1 metre to the radius must require adding kilometres of rope. The algebra disagrees completely.
Let's look at the mathematical proof:
1. Let the original radius of the circle be R. The original circumference is C1 = 2πR.
2. When the rope is raised by a gap of h (where h = 1 metre), the new radius becomes R + h.
3. The new circumference is C2 = 2π(R + h) = 2πR + 2πh.
4. The extra rope needed is the difference between the new and old circumferences:
Extra = C2 - C1 = (2πR + 2πh) - 2πR = 2πh.
Notice that the variable R (representing the radius of the Earth) cancels out of the equation entirely! This means that the amount of extra rope required depends solely on the height of the gap (h), not the size of the body being encircled.
To explain the WOW part:
- For a small marble (R = 1 cm): raising the rope by 1 metre requires 2π × 1 = 6.28 metres of extra rope.
- For the Earth (R = 6,371 km): raising the rope by 1 metre requires 2π × 1 = 6.28 metres of extra rope.
- For the Sun (R = 696,340 km): raising the rope by 1 metre requires 2π × 1 = 6.28 metres of extra rope.
Because the derivative of a circle's circumference with respect to its radius is a constant (dC/dR = 2π), the rate of circumference growth is perfectly linear and independent of scale.