course Mth 152 ÿ…úœòÿò²EêжÜÃبšÑ™assignment #025
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22:47:24 **** query 9.7.6 intersecting lines m, n parallel to k **** In which geometry or geometries is this possible and why?
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RESPONSE --> Riemannian
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22:47:46 ** Two parallel lines intersect on a sphere (think of lines of longitude). So this occurs in a Riemannian geometry. **
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RESPONSE --> Ok.
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22:48:10 **** query 9.7.18 ruler r.b. CD wrench nail **** To which of the objects is the coin topologically equivalent and why?
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RESPONSE --> Wrench and ruler
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22:48:16 The coin is topologically equivalent to the ruler and the wrench nail because none of these have holes.
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RESPONSE --> Ok.
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22:48:51 **** query 9.7.27 genus of 3-hole-punched sheet of paper **** What is the genus of the sheet and why?
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RESPONSE --> Genus of 3 since it has 3 holes
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22:49:04 The genus of this sheet of paper is 3 becasue it contains 3 holes.
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RESPONSE --> Ok.
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22:50:45 **** query 9.7.42 3,3,3,3,4,4,2,2 **** Can the network be traversed or not and why?
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RESPONSE --> No, a network with 4 vertices can't be traversed.
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22:50:52 ** This network contains 4 odd vertices. A network with 0 or 2 odd vertices can be traversed; a network with 4 odd vertices cannot. **
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RESPONSE --> Ok.
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22:54:10 **** If you start on a vertex of order 3 can you traverse the network and end up on that vertex? Explain why your answer must be true.
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RESPONSE --> No, when you cross the second edge you are back to the vertex and when you go to the third you can't get back to it.
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22:54:21 ** You can’t start on a vertex of order 3 and end up on the same one. You leave the vertex along the first of the three edges. When you traverse the second of these edges you are returning to the vertex, and when you leave again you have to travel along the third and you can't get back. You can end up on a different vertex of degree 3 if there is one (and if there is one you must end on it), but you can't end up on the degree-3 vertex you started from. **
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RESPONSE --> Ok.
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22:55:47 **** If you start on a vertex of order 4 can you traverse the network and end up not on that vertex? Explain why your answer must be true.
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RESPONSE --> No, you would go back to the vertex on the fourth edge.
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22:56:15 ** If you start on a vertex of order 4 you cannot traverse the network without ending up on that vertex, since you leave the vertex on the first edge, return on the second and leave on the third. If you traverse the network you have to return to the vertex on the fourth edge, and you can’t leave again. **
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RESPONSE --> Ok.
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22:56:49 **** If you start on a vertex of order 2 and traverse the network must you end up on that vertex? Explain why your answer must be true.
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RESPONSE --> You leave on the first and return on the second.
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22:57:12 ** If you start on a vertex of order 2 and traverse the network you leave on the first edge, return on the 2 nd and you’re stuck there. **
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RESPONSE --> Ok.
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22:57:23 No, because once again this is an even vertex. One point must be the starting point and one the ending point.
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RESPONSE --> Ok.
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23:00:26 **** If you start off of a certain vertex of order 3 and traverse the network is it possible to end up somewhere besides this vertex?
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RESPONSE --> No, you go on the first and come back on the second. So when you leave on the third you're stuck.
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23:01:13 ** If you start on a vertex of order 3 and traverse the network you leave on the first, return on the second and leave on the third edge. You can’t travel any of these edges again so you can never return. Therefore you must end up elsewhere. **
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RESPONSE --> Ok.
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23:01:40 **** Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> None.
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|~ìy÷¹…z±²£¹ê쟩{¬ÃÃìàö¢dÉž¢ assignment #026 026. `query 26 Liberal Arts Mathematics II 08-01-2007
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23:09:36 **** query 9.8.9 table of size ratio to scale factor for squares **** What are the size ratios for scale factors 2, 3, 4, 5, 6 and 10, and what is the relationship between size ratio and scale factor?
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RESPONSE --> 2 = 4 3 = 9 4 = 16 5 = 25 6 = 36 10 = 100 Size ratio is the square of the scale factor.
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23:10:07 The size ratios for scale factor 2 is 4, for 3 it is 9, 4 is 16, 5 is 25, 6 is 36, 10 is 100. The size ratio is the square of the scale factor.
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RESPONSE --> Ok.
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23:16:04 17:44:38
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RESPONSE -->
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23:16:22 In order to form a larger square by adding to the smaller one it must have the same number of edges across the top, bottom, and both sides to stay square. The size or the area of the square is found by multplying the length times the width. A square's length is the same as its width, so all you are doing is squaring one side to get the size.
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RESPONSE --> Ok.
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23:17:46 **** query 9.8.15 putting unit cubes together to make next larger cube **** What are the scale factor and size ratio for the two cubes?
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RESPONSE --> Scale Factor: 2 Size Ratio: 8
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23:18:00 The scale factor is 2 and the size ratio is 8.
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RESPONSE --> Ok.
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23:20:32 **** query 9.8.18 dimension of cube **** What is the dimension of a cube?
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RESPONSE --> L*W*H
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23:20:49 The dimension of a cube is the length times the width times the height. A cube with a scale factor of 2 would have a size ratio of 2 * 2 * 2 or 2^3.
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RESPONSE --> Ok.
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23:22:59 **** How does the relationship between size factor and scale factor tell you that the cube is 3-dimensional?
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RESPONSE --> You cube the scale factor to find the size ratio.
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23:23:14 Because you have to cube the scale factor to get the size ratio.
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RESPONSE --> Ok.
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23:32:46 **** query 9.8.24 Sierpinski gasket **** What are the length factor and the size factor for this figure, and what two whole numbers therefore must its dimensions therefore lie between?
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RESPONSE --> Length Factor: 2 Size Factor: 3 Dimensions between 1 and 2.
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23:33:56 ** If you double the length you get 3 additional copies of the original figure. So length factor and size factor are 2 and 3. The dimension is the number such that (scale factor) ^ dimension = size factor. For example for the cube a doubling of scale factor increased size factor to 8 times its original value. This gives us the equation 2^d = 8, and as we saw above d = 3 for a cube. Here scale factor is 2 and size factor is 3 so we need to find d such that 2^d = 3. Since 2^1 = 2 and 2^2 = 4, d must be between 1 and 2. **
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RESPONSE --> Ok.
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23:34:26 **** When you double the scale of the gasket by doubling its width, how many new copies of the original figure do you get?
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RESPONSE --> 3
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23:34:49 ** You get 3 copies of the original figure. **
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RESPONSE --> Ok.
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23:35:58 **** Since a doubling of a scale increases size by factor 3, is the dimension greater or less than 1, and is the dimension greater or less than 2?
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RESPONSE --> Greater than 1, less than 2.
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23:36:21 *&*& If the dimension was 1 then doubling the scale would double the size. If the dimension was 2 then doubling the scale would give you 2^2 = 4 times the size. Since doubling gives you 3 times the size, the dimension must be greater than 1 and less than 2. *&*&
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RESPONSE --> Ok.
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23:36:53 **** What equation would you solve to get the dimension?
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RESPONSE --> 2^D = 3
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23:37:13 ** The equation (see above note) is 2^d = 3. The solution is about d = 1.59, as you say below. **
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RESPONSE --> Ok.
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23:38:10 **** Note that the equation is 2^d = 3. What approximate value of d makes this equation true?
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RESPONSE --> D = 1.59
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23:38:51 *&*& By trial and error we find that d = 1.585 comes close to making this equation true. *&*&
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RESPONSE --> Ok.
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23:39:07 **** Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> None.
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