The “rope around the Earth” problem

My son ran into a version of the classic “rope around the Earth” problem yesterday. The common version of the problem goes something like this:

You have a rope that goes all of the way around the Earth at the equator. If you wanted a second rope to also go all the way around the Earth but always be exactly 1 ft higher up than the first rope, how many feet longer than the first rope would the second rope need to be? (Assume that the equator is a perfect circle).

The version of the problem that my son saw was slightly different. I thought it would be fun to use that problem to motivate a short project about perimeter and area of some simple shapes:

The first step in our exploration was adding one blue strut to each side of the shapes. How did the area change? How did the perimeter change?

It was easier for my older son to see the change in area in his square than it was for my younger son to see the change in his triangle. However, after talking for a little bit, we were able to resolve the discrepancy.

The next step was looking at the pattern in the area and perimeter when we added one more blue strut to the sides of each shape. At this stage the boys seem to have a pretty good handle on how the area and perimeters are changing and even were able to make a conjecture about the next step.

In the last part of today’s project we take a look to see if the patterns that the boys thought would emerge on the last step do indeed emerge. We then turn back to the original “rope around the Earth” problem to see how to understand the different perimeters in play there.

So, a fun project motivated by a classic problem. Hopefully the talk allowed the boys to see that there more than just a clever little problem going on here.

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