Measuring Distant Objects

Transcription

1 Measuring Distant Objects 17 March 2014 Measuring Distant Objects 17 March /30

2 How can you measure the length of an object? Ideally, you use a ruler, a tape measure, or some other measuring device. But what if you can t get to the object to physically measure it? For instance, what if it is the height of a mountain? This week we will explore how certain objects can be measured. Today we will focus on terrestrial objects, such as pyramids or buildings. Later this week we will look at how the distance to the earth and moon, and the size of the earth, was approximated over 2000 years ago. Measuring Distant Objects 17 March /30

3 Thales of Miletus Thales, born around 624 B.C., was a pre-socratic Greek philosopher. Many, including Aristotle, regard him as the first philosopher in the Greek tradition. Thales rejection of mythological explanations became an essential idea for the scientific revolution. He was also the first to define general principles and set forth hypotheses, and as a result has been dubbed the Father of Science. One of the mathematical problems Thales solved was how to measure the heights of the Egyptian pyramids. Measuring Distant Objects 17 March /30

4 Great Pyramid of Cheops Measuring Distant Objects 17 March /30

5 Clicker Question Could we take a long rope to the top and measure how much of the rope it takes to reach the bottom? A Yes that should work. B No it won t give the right height. Measuring Distant Objects 17 March /30

6 Answer Unfortunately, that won t work. It would give the length of a diagonal side of the pyramid. That is longer than the height of the pyramid. If we could cut a hole in the pyramid straight down and drop the rope until it hits the ground, measuring the length of the rope would give us the height. Measuring Distant Objects 17 March /30

7 Illustration of a Variant of Thales Method Measuring Distant Objects 17 March /30

8 What did Thales Do to Measure the Height? Thales discovered a way to measure the height of the pyramids. Thales reasoned that if his height was the same as the length of his shadow, then the same should be true for the pyramid. He waited till his height equaled his shadow, then measured the shadow of the pyramid. From this he knew the height of the pyramid. Measuring Distant Objects 17 March /30

9 Thales Discovery as a Beginning of Mathematics Auguste Comte: In light of previous experience we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know... In renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. Auguste Comte 19th century philosopher Measuring Distant Objects 17 March /30

10 At the heart of Thales discovery is the notion of proportionality. Plutarch, a Greek historian, gives another version of Thales measurement: The height of a pyramid is related to the length of its shadow just as the height of any vertical object is related to the length of its shadow at the same time of day. This is more powerful than what Thales did. We ll discuss the idea behind Plutarch s statement in some detail, and make it precise. Measuring Distant Objects 17 March /30

11 Proportionality and Scaling Measuring Distant Objects 17 March /30

12 What happens when you put an image in a copy machine and enlarge or shrink the image? The pictures on the previous page are the same, except that the right-hand picture is enlarged to 200% of the left-hand picture. By doubling the size of the picture, each length gets doubled. If King Kong was 2 inches tall in the first picture, he d be 4 inches tall in the second picture. Measuring Distant Objects 17 March /30

13 Clicker Question If you double King Kong s dimensions, do you think his weight doubles? A Yes B No Measuring Distant Objects 17 March /30

14 Answer His weight would increase much more than twice. One way to think about this is to think of a cube of some material. If each side length doubles, then the volume increases by a factor of 8. The weight would increase by a factor of 8. It turns out that this is the reason why King Kong and flies the size of humans are fictional. Bones aren t strong enough to handle the increased weight. But, that is a story for another time. Measuring Distant Objects 17 March /30

15 Similar Triangles If we take one triangle and enlarge or shrink it, as a copy machine would do, we get another triangle. These two triangles are called similar. Each angle of the small triangle is equal to one of the angles in the big triangle. The equal angles are marked with the same color. Measuring Distant Objects 17 March /30

16 When the angles of one triangle are equal to the angles of another, then one triangle is obtained from the other by magnifying (or shrinking), and the two are similar. If we think of taking the smaller figure and magnifying it with a copy machine, then the scale factor represents how much we magnify. For example 200% corresponds to a scale factor of 2, and 150% corresponds to a scale factor of 1.5. The scale factor says by what factor each length grows when going from the smaller figure to the larger figure. Measuring Distant Objects 17 March /30

17 In the following picture, the right-hand picture was made by taking the left-hand picture and magnifying it by 200%. Note that the two line segments have doubled length but the size of the angle is the same. Scaling does not change angles. Measuring Distant Objects 17 March /30

18 The length of the segment EG can be found by multiplying the length of AB times the scale factor. Similarly, the length of EF is the length of AC times the scale factor, and similarly for the third sides. Written as equations, if s is the scale factor, then EG = s AB EF = s AC GF = s BC Measuring Distant Objects 17 March /30

19 By dividing, we can write these as EG AB = s EF AC = s GF BC = s We can write this without reference to the scale factor as EG AB = EF AC = GF BC This is a useful set of equations coming from similar triangles. Measuring Distant Objects 17 March /30

20 In words, this relationship can be described as: If two triangles are similar, then corresponding sides are proportional, meaning that the ratio of the lengths of corresponding sides is the same. Corresponding sides represent an original side and a scaled side. For example, AB and EG are corresponding sides. Corresponding angles are those drawn in the same color above. Measuring Distant Objects 17 March /30

21 Clicker Question These two triangles are similar. How long is the unknown side? Measuring Distant Objects 17 March /30

22 Answer The length is 2 inches. It is the solution of the equation??? 1 = = 2 Another way to answer this is to note that, due to the top sides, the scale factor is 3/1.5 = 2. Thus, we have to multiply each length of the left-hand triangle by 2 to get the corresponding length in the right-hand triangle. Therefore, the unknown length is 2 inches. Measuring Distant Objects 17 March /30

23 Proportionality The relationship between corresponding sides of similar triangles is an example of a proportional relationship. Two variable quantities are said to be proportional if their ratio is a constant. Another example is the ratio of height to arm length in any photo of King Kong. No matter how much we scale the picture, the ratio will be the same. If, say we scale the picture by 200%, King Kong s height will double, but so will his arm length. So, the ratio of height to arm length will remain the same. Another example comes from circles. The ratio of circumference to diameter (twice the radius) is always constant. The ratio is the number π. Measuring Distant Objects 17 March /30

24 Plutarch s Variant of Thales Method Thales understood similar triangles, and how similar triangles could be used to measure heights. Measuring Distant Objects 17 March /30

25 In this picture the two triangles are similar because they have equal corresponding angles. The blue angles are equal because the sun s rays are parallel. Measuring Distant Objects 17 March /30

26 Because these triangles are similar, corresponding sides are proportional. This means height of clock tower height of stick = length of clock tower shadow length of stick shadow In this equation, we know or can measure three things, the two shadows and the height of the stick. We can then solve for the height of the clock tower. Measuring Distant Objects 17 March /30

27 An Example Suppose the stick is 3 feet high, it casts a shadow of 2 feet, and the clock tower casts a shadow of 25 feet. Our equation then becomes height of clock tower height of stick = height of the clock tower 3 length of clock tower shadow length of stick shadow = 25 2 We can multiply by 3 to get height of the clock tower = = 75 2 = 37.5 feet Measuring Distant Objects 17 March /30

28 The benefit of using similar triangles is that we don t have to wait for our shadow (or that of the stick) to be the same length as our height, as Thales did. We can do this measurement at any point in time. One drawback to this method is that we need to have a sunny day, so that we can see shadows. Another drawback is that we may not be able to measure a shadow. We need to have space around us in order to measure the shadow. This wouldn t be feasible in many situations. For instance, we couldn t measure the height of Organ Peak in this way. Measuring Distant Objects 17 March /30

29 Next Time We will explore some methods that will overcome the problems mentioned in the last slide. Each of the methods we consider will use similar triangles. Measuring Distant Objects 17 March /30