.
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.
--------------
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.