Volume of a Cone, Part III
This will now be the third post in a row on the volume of a cone that doesn't mention cones or how to generate them. Stay with me. We'll get there.
In Part I, we noted that we can generate a cylinder using a rotating rectangle. In Part II, we saw how to use that rotating rectangle and the idea of average to come up with a formula for the volume of a cylinder.
Last time, I used an example of runners on a track. If we have seven runners, all running different distances—say, 12 miles, 10 miles, 8 miles, 6 miles, 4 miles, 2 miles and 0 miles (for a total of 42 miles)—the average distance they run (6 miles) tells us the one distance each runner could have run in order to cover the same total distance (6 miles × 7 runners = 42 miles).
With the rotating rectangle, we saw that all the points in the area of the rectangle (rh) were moving different distances (for a total distance that we didn't know). The average distance they moved, 2&pi(1/2r), based on their average distance from the center, told us the one distance every point could have moved in order to cover the same total distance.
Well, we can take this a step further. Take a look at the image below. It shows the weights and ages of, say, 49 cats. Each cat (represented by a point) has one of seven ages (in years) and one of seven weights (in pounds). [I know, it's lame.]
If we want to find the average weight of the cats in the survey, we can add all the weights (7 × 0 + 7 × 1 + 7 × 2 + 7 × 3 + 7 × 4 + 7 × 5 + 7 × 6 = 7(0 + 1 + 2 + 3 + 4 + 5 + 6) = 7 × 21 = 147) and divide that number by the number of cats (49) to come up with our average (147 ÷ 49 = 3).
Or, we can find the average weight for each age--from 0 to 6—to begin to find the average weight for all the cats. The average weight for each age is the same ([0 + 1 + 2 + 3 + 4 + 5 + 6] ÷ 7 = 3).
This is exactly what we did with the rotating rectangle in the previous post. The "average" of every horizontal width of the rectangle could be found by drawing a line down the center of every horizontal width. Since every horizontal width was the same length, they all had the same "average," meaning that, for our purposes, all the points of the rectangle could be described as lying along that vertical center line of the rectangle. And that's all we needed to know to find the volume of the cylinder.
But, going back to our survey, let's take a look at the average age for each weight. Obviously, for these data, the average is going to be the same, 3.
The image below shows both of these averages—for age and weight. You can see that there are seven cats with the average weight and seven cats with the average age. And, lo and behold, there is actually one kitty (the green point) that has both the average weight AND the average age.
Thus, all of the data above can be "described" using just that one point. That is, if every cat in the group had the same age AND the same weight, they would all have the same age and weight as that green cat in the center. (And the total weight of all the cats and the total age of all the cats would stay the same.)
For our rotating rectangle, not only can we draw a line bisecting every horizontal line segment, but we can draw a horizontal line bisecting every vertical line segment. The point where these two lines cross (1/2r, 1/2h) can be used to "describe" every point in the area of the rectangle. This point is called the centroid of the rectangle.
Now, as I alluded to before, finding the centroid of the rectangle wasn't necessary to come up with the volume formula. We simply found that we could describe every point as though they all lay on the vertical center line at a distance of 1/2r from the center.
But imagine, for example, that our age-weight data covered only 28 cats and looked like this:
If we find the average weight for each age (left image below) and the average age for each weight (right image below), we still get a line each time as we did with the previous data, but, in this case, neither of those lines runs through just one number.
So it is important to see where these lines intersect. As it turns out, they intersect at (2, 2), making this point the centroid.
When we add all the ages (56) and divide by the number of cats (28), we get 2. And when we add all the weights (56) and divide by the number of cats (28), we also get 2. Thus, if every cat weighed the same and was the same age, they'd be two years old and weigh two pounds, and we would still get the same total weight and the same total age.
Next time, we'll finally get to see how all of this applies to the volume of a cone.
Volume of a Cone: Part I | Part II | Part III | Part IV
In Part I, we noted that we can generate a cylinder using a rotating rectangle. In Part II, we saw how to use that rotating rectangle and the idea of average to come up with a formula for the volume of a cylinder.
Last time, I used an example of runners on a track. If we have seven runners, all running different distances—say, 12 miles, 10 miles, 8 miles, 6 miles, 4 miles, 2 miles and 0 miles (for a total of 42 miles)—the average distance they run (6 miles) tells us the one distance each runner could have run in order to cover the same total distance (6 miles × 7 runners = 42 miles).
With the rotating rectangle, we saw that all the points in the area of the rectangle (rh) were moving different distances (for a total distance that we didn't know). The average distance they moved, 2&pi(1/2r), based on their average distance from the center, told us the one distance every point could have moved in order to cover the same total distance.
Well, we can take this a step further. Take a look at the image below. It shows the weights and ages of, say, 49 cats. Each cat (represented by a point) has one of seven ages (in years) and one of seven weights (in pounds). [I know, it's lame.]
If we want to find the average weight of the cats in the survey, we can add all the weights (7 × 0 + 7 × 1 + 7 × 2 + 7 × 3 + 7 × 4 + 7 × 5 + 7 × 6 = 7(0 + 1 + 2 + 3 + 4 + 5 + 6) = 7 × 21 = 147) and divide that number by the number of cats (49) to come up with our average (147 ÷ 49 = 3).
Or, we can find the average weight for each age--from 0 to 6—to begin to find the average weight for all the cats. The average weight for each age is the same ([0 + 1 + 2 + 3 + 4 + 5 + 6] ÷ 7 = 3).
This is exactly what we did with the rotating rectangle in the previous post. The "average" of every horizontal width of the rectangle could be found by drawing a line down the center of every horizontal width. Since every horizontal width was the same length, they all had the same "average," meaning that, for our purposes, all the points of the rectangle could be described as lying along that vertical center line of the rectangle. And that's all we needed to know to find the volume of the cylinder.
But, going back to our survey, let's take a look at the average age for each weight. Obviously, for these data, the average is going to be the same, 3.
The image below shows both of these averages—for age and weight. You can see that there are seven cats with the average weight and seven cats with the average age. And, lo and behold, there is actually one kitty (the green point) that has both the average weight AND the average age.
Thus, all of the data above can be "described" using just that one point. That is, if every cat in the group had the same age AND the same weight, they would all have the same age and weight as that green cat in the center. (And the total weight of all the cats and the total age of all the cats would stay the same.)
For our rotating rectangle, not only can we draw a line bisecting every horizontal line segment, but we can draw a horizontal line bisecting every vertical line segment. The point where these two lines cross (1/2r, 1/2h) can be used to "describe" every point in the area of the rectangle. This point is called the centroid of the rectangle.
Now, as I alluded to before, finding the centroid of the rectangle wasn't necessary to come up with the volume formula. We simply found that we could describe every point as though they all lay on the vertical center line at a distance of 1/2r from the center.
But imagine, for example, that our age-weight data covered only 28 cats and looked like this:
If we find the average weight for each age (left image below) and the average age for each weight (right image below), we still get a line each time as we did with the previous data, but, in this case, neither of those lines runs through just one number.
So it is important to see where these lines intersect. As it turns out, they intersect at (2, 2), making this point the centroid.
When we add all the ages (56) and divide by the number of cats (28), we get 2. And when we add all the weights (56) and divide by the number of cats (28), we also get 2. Thus, if every cat weighed the same and was the same age, they'd be two years old and weigh two pounds, and we would still get the same total weight and the same total age.
Next time, we'll finally get to see how all of this applies to the volume of a cone.
Volume of a Cone: Part I | Part II | Part III | Part IV
Labels: mathematics
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