Riddler Express 8/14/2020

Original post here (fivethirtyeight.com).

From Dean Ballard comes a sneaky sorting of squares:

Suppose you have a rope that goes all the way around the Earth’s equator, flat on the ground. (For the entirety of this puzzle, you should assume that the Earth is a perfect sphere with a radius of 6,378 kilometers.)

You want to lengthen the rope just the right amount so that it’s 1 meter off the ground all the way around the Earth. How much longer did you have to make the rope?

If you’ve never heard this puzzle before, the answer is surprisingly small — about 6.28 (i.e., 2𝜋) meters. Also, spoiler alert! (Darn, I was one sentence too late.)

Now, instead of tying the Earth up with rope, you’ve moved on to covering the globe with a giant sheet that lies flat on the ground. If you want the sheet to be 1 meter off the ground (just like the rope), by how much would you have to increase the area of your sheet?

Extra credit: What city, country, land mass or body of water has an area that is very close to your answer?

Answer

51.028π (approximately 160.309) additional square kilometers.

Derivation

Setting up the problem, we define:

We relate them:

2

We also define r1 to be the radius of the original globe, and r2 to be the radius of the sheet. From the definition of the problem, we know:

3

Next, recall the formula for the surface area of a sphere:

1

Since both the original globe and the sheet are a sphere (in this problem, anyway), we have

4

In the problem, we’re told to assume that the radius r1 of the globe is 6378 km. Therefore, r2 = 6379 km. Substituting those into the previous equations, we have:

5

Then, we simplify algebraically and solve for x:

6

7

8

9

10

Note that there are 1,000,000 square meters (not 1,000) in 1 square kilometer! Therefore, we have to divide by 1,000 to get the actual answer.

Commentary

But wait, why is the additional surface area required for the sheet (51028π) so much larger than the additional length required for the rope (2π)?

The derivation for the sheet above shows that the additional surface area required is proportional to the difference of the square of the globe and sheet’s radii.

11

However, the additional length required for the rope doesn’t depend on the radius at all. To see why, recall the formula for the circumference of a circle (i.e. the length of the rope on the ground):

12

To find the length of the rope 1m above the surface, we can use a similar setup as we did above for the sheet example:

13

14

Substituting and solving for x, we get:

15

16

17

The idea of a quantity growing rapidly as dimensions are added is common across a number of mathematical disciplines. For more on this idea, see on the curse of dimensionality.