Volume of a Cone, Part II
In Part I, we took a look at an interesting problem—how to find the volume formula for a cylinder by rotating a rectangle about one of its sides.
The tricky part, as I mentioned, is that not every point in the area of the rotating rectangle moves the same distance, so we can't simply take the area of the rectangle (rh) and multiply it by some distance.
Well, actually we CAN do this, but we need to do some reasoning before we can SEE that we can do this. And that reasoning, as I mentioned before, involves not geometry, but elementary data analysis. In particular, the idea of average.
If we think of each number shown as a radius (and we remember that the formula for the circumference of a circle is 2&pir), then we know that the runner standing in Lane 3 will run a distance of 2&pi(3), the runner standing in Lane 1 will run a distance of 2&pi(1), etc. In order from least to greatest, the distances the runners will run are 0, 2&pi, 4&pi, 6&pi, 8&pi, 10&pi, and 12&pi .
How far will ALL the runners run? That is, if one person ran 12&pi in a straight line, then tagged the next person who ran 10&pi, etc., what distance would they cover? Of course, we could simply add up the distances to get our answer: 42&pi .
Another way to find the total distance the runners will run is to find the average distance they will run and multiply that by the number of runners. And since the distance each runner will run is dependent on his or her distance from the center, we can simply find the average distance the runners are from the center, and then multiply that by 2&pi and by the number of runners.
Obviously, the average distance the runners are from the center is 3 ([0 + 1 + 2 + 3 + 4 + 5 + 6] ÷ 7 = 3). Multiply that by 2&pi (2&pi × 3 = 6&pi). Then multiply that by the number of runners (6&pi × 7 = 42&pi).
By calculating the total distance using the average distance the runners are from the center, we have shown that the total distance the runners will run if they each run in their separate lanes is the same as the total distance they would run if they were all in Lane 3.
Bisecting all of these horizontal widths gives us the "average" of each width—1/2r--meaning that the average distance every point is from the center of the circular base is 1/2r. And, because we used the average distance all the points are from the center of the circular base (1/2r), we can see that the total distance all the points of the rectangle will move if they move at different distances from the center is the same as the total distance they would move if they were all 1/2r from the center.
So, we take the area of the rectangle (rh)--or, all of its points--and multiply that by the distance all of those points would move, 2&pi(1/2r). The result, of course, is &pir2h, the volume of the cylinder.
We can use (almost) this same method to find the volume of a cone. Coming soon!
Volume of a Cone: Part I | Part II | Part III | Part IV
The tricky part, as I mentioned, is that not every point in the area of the rotating rectangle moves the same distance, so we can't simply take the area of the rectangle (rh) and multiply it by some distance.Well, actually we CAN do this, but we need to do some reasoning before we can SEE that we can do this. And that reasoning, as I mentioned before, involves not geometry, but elementary data analysis. In particular, the idea of average.
Finish Your Radius, Or You Can Forget About Pi
The image below shows a rather strange circular running track with six different circular lanes. The points represent seven different runners—one standing at the very center of the track (0), one standing at a distance of 1 from the center, another standing at a distance of 2 from center, and so on. Each runner will run one time around the track in his or her lane—except for the lucky guy or gal standing at the center; he or she won't have to move.If we think of each number shown as a radius (and we remember that the formula for the circumference of a circle is 2&pir), then we know that the runner standing in Lane 3 will run a distance of 2&pi(3), the runner standing in Lane 1 will run a distance of 2&pi(1), etc. In order from least to greatest, the distances the runners will run are 0, 2&pi, 4&pi, 6&pi, 8&pi, 10&pi, and 12&pi .
How far will ALL the runners run? That is, if one person ran 12&pi in a straight line, then tagged the next person who ran 10&pi, etc., what distance would they cover? Of course, we could simply add up the distances to get our answer: 42&pi .
Another way to find the total distance the runners will run is to find the average distance they will run and multiply that by the number of runners. And since the distance each runner will run is dependent on his or her distance from the center, we can simply find the average distance the runners are from the center, and then multiply that by 2&pi and by the number of runners.
Obviously, the average distance the runners are from the center is 3 ([0 + 1 + 2 + 3 + 4 + 5 + 6] ÷ 7 = 3). Multiply that by 2&pi (2&pi × 3 = 6&pi). Then multiply that by the number of runners (6&pi × 7 = 42&pi).
By calculating the total distance using the average distance the runners are from the center, we have shown that the total distance the runners will run if they each run in their separate lanes is the same as the total distance they would run if they were all in Lane 3.
The Rest Is Gravy
We can think of a cylinder as an infinite collection of these circular tracks and the radii (r) of those tracks as all the horizontal widths (i.e., the entire area) of the rectangle we will use to generate the cylinder. There are an infinite number of runners (meaning we don't care how many there are) represented by points along each radius.Bisecting all of these horizontal widths gives us the "average" of each width—1/2r--meaning that the average distance every point is from the center of the circular base is 1/2r. And, because we used the average distance all the points are from the center of the circular base (1/2r), we can see that the total distance all the points of the rectangle will move if they move at different distances from the center is the same as the total distance they would move if they were all 1/2r from the center.So, we take the area of the rectangle (rh)--or, all of its points--and multiply that by the distance all of those points would move, 2&pi(1/2r). The result, of course, is &pir2h, the volume of the cylinder.
We can use (almost) this same method to find the volume of a cone. Coming soon!
Volume of a Cone: Part I | Part II | Part III | Part IV
Labels: mathematics