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Question 1152563: Imagine that a rope is wrapped tightly around the earth at the equator. Then suppose that the cable is cut, an additional 20 feet are added to the rope, and that this longer rope is put back around the equator, but now it is hovering around the earth with equal space between the rope and the equator. Because the rope is now a little longer than the circumference of the earth, there will be a gap between the rope and the surface of the earth. How large is the gap?
Items for Discussion
● What does this look like? (make sure you and your partner understand the problem, it may help to draw a picture)
● How big do you think the gap is? What’s your guess?
● Do you think you would be able to walk under the rope? Crawl under the rope? Slide your hand under the rope?
The diameter of the Earth at the equator is about 7,926 miles.
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617)   (Show Source):
Answer by ikleyn(52777)  (Show Source):
You can put this solution on YOUR website! .
This problem is a classic entertainment problem.
Each mathematically educated person must solve it several times in his (or her) life:
first time for himself / herself at the age from 9 to 12;
second time with his or her children;
and the third time with his or her grandchild.
But, surely, it should be solved correctly and presented as a real entertainment problem -- otherwise, what is the reason to do it, at all ?
The formulas in the post by @josgarithmetic are written INCORRECTLY, and you will NEVER get a right answer using them.
Therefore, I came to do the job for you in a way AS IT SHOULD BE done.
The diameter of the Earth at the equator is about 7,926 miles;
hence, the radius of the Earth is  = 3963 miles, or about 3963*5280 feet = 20924640 feet.
The circumference of the Earth, at given radius is
C =  feet = 2*3.14159265*20924640 feet = 131473390.5 feet.
It is the original length of the rope.
After adding 20 feet, the length of the rope is 131473390.5 + 20 = 131473410.5 feet.
The new radius R satisfies this equation for the circle circumference
 = 131473410.5,
which gives you THIS value for R
R =  =  = 20924643.2 feet.
The difference 20924643.2 - 20924640 = 3.2 feet is exactly the value of the gap.
I do not know which height are you.
But the child whose height is less than 3 feet, definitely could walk under the rope.
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The common expectation of a person, unfamiliar with this phenomenon, is that the gap should be extremely small.
The amazing fact is that IT IS NOT so.
In reality, the gap would be sufficient for a dog or for a goat to walk under the rope.
In this case, elementary Mathematics demonstrates its power in analyzing such an amazing fact.
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