PUZZLE
# 013
June 20, 2016
PUZZLE
# 013
Imagine the Earth as a perfect sphere. You tie a rope
around the equator of the Earth, so that the rope is
(approximately) 40 000 000 m long. The rope goes all
the way around the Earth and meets itself, so that it is
the exact length of the circumference (Fig. 1).
A LONG
ROPE
The Question: If you add just 2 m to the rope, and
then lift the entire rope up from the surface, so that it
is still in a perfect circle and the same distance off the
ground all the way around, how high off the ground is
the rope (Fig. 2 & 3)?
METHOD
PROCESS
1
How We Can Look At The Problem:
The goal is to figure out the distance (x) from the
surface of the Earth to the rope (Fig. 3). This distance
can be considered as the Earth’s diameter plus 2x to
find the diameter of the new rope circle.
CEarth = 40 000 000 m
Rope around
equator
Fig 1: Rope around equator of the Earth
What We Know:
Circumference is calculated by C = πd
We know that the circumference of the lifted rope is
CEarth+ 2 m (Fig. 2).
We also know that the diameter is dEarth + 2x: where x is
the distance that the rope is lifted off of the surface.
2m
Fig 2: Adding an extra 2 m to the rope
What Is The Equation:
2
We can start by coming up with an equation for the
circumference of the extended rope:
x
CEarth + 2 m = π(dEarth + 2x)
We need 2x added to the diameter to account for
the fact that the rope is lifted an equal amount from
the surface all the way around the Earth. So, x will be
accounted for twice, one on either side of the Earth
(Fig. 4).
What We Can Compare:
Fig 3: Rope equally lifted off the surface
The circumference of the extended rope can be
compared with the circumference of the original:
3
CEarth = πdEarth
Since we are looking for the difference in the
diameter of the two cases, we can subtract this
equation from the equation for the extended rope.
(CEarth + 2 m = πdEarth + 2πx) - (CEarth = πdEarth)
x
dEarth = 12 742 000 m
2m = 2πx
Therefore, the rope can be lifted equally: 1m/π or
0.318 m off the surface of the Earth.
SOLUTION
A rope that is 2 m longer than the circumference of
the Earth can be lifted an equal distance of 0.318 m
above the ground.
For every 1 m added to the length of the rope, it can
be lifted 0.159 m, or 15.9% of the length added.
CEarth =40 000 000 m
Fig 4: Distance x from Earth’s surface
x