Rappeling Down Fissures

This past weekend I went to Yosemite and bought an excellent guidebook to hike the park. If you haven't been to Yosemite, you really must go. John Muir, the premier naturalist of American history, once said that Yosemite Valley was the most glorious temple he of Nature he had ever been lucky enough to see. That part of the Sierra Nevada mountain range is so different than almost any others in the world. As is mentioned in the guide book, the park itself ranges in elevation from 2000 ft to over 13000 ft, an amazing 11,000 ft vertical range over 1200 square miles.



This guidebook mentions a place that I would like to go to soon, called Taft Point and the Fissures, . Apparently there are fissures in some rocks in the high country where you can look down THOUSANDS of feet to the valley floor. That is amazing. I hope to have my own picture soon, but the one here I found on the web.

This morning I showed this to one of my students, and he thought that it would be a good idea to rappel down the inside of these fissures. I, deciding to crush his imaginative spirit in the name of Truth, told him that it might require a prohibitive amount of rope to rappel down that long of a distance.

And, much to his chagrin, that we could use math to figure out whether it would even be possible.


So HOW would we go about figuring out if we could carry enough rope to rappel clearly down the fissure from the top? We might begin by asking how much space that rope would take up, known as its volume, and see if we could fit it into a pack. We may also want to know how heavy that much rope would be, or its weight, to see if a normal human being could carry that much rope with us.

My student and I decided to do this calculation for rappeling down a 3000 ft rock, using a rope that is 1.5" thick. We further assumed that the rope could be modelled as a really long thin cylinder, allowing us to use the volume formula for a cylinder. And since most backpack sizes are given in cubic inches, and not cubic feet, we decided to report the total volume of the rope in cubic inches.

This problem was interesting as I was trying to explain how it is that we can derive the formula for the volume of a solid like a cylinder or a cube, but can't so easily with a more complicated shape like a sphere or a pyramid.

DIMENSIONS

In understanding geometric shapes, it is helpful to think about the idea of dimensions. Most people have a good intuitive sense of what is meant by one, two and three dimensions. A one dimensional object would be a line, a two dimensional object would be something that you could draw on a piece of paper, and a three dimensional object would pretty much describe most solid geometric objects that we encounter in our experiences.

The nice thing about certain types of three dimensional geometric objects, such as cubes, cylinders and other objects that have uniform cross sections (e.g. their cross section is the same everywhere along their height), is that we can extend our formula for the area of the cross section to a formula for the volume of the solid object almost immediately. Before we do so, it is helpful to refresh our memory for calculating the area of two dimensional objects.