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Unformatted text preview: (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 (f) 5 Solution Key: 2.8 Solution: 3.8 Question 9: Label the consequtive edges of an Euler circuit in the following graph: Solution Key: 2.9 Solution: 3.9 8 Question 10: True or False. Give a reason or a counterexample. (1) There exists a planar graph with 6 vertices, 10 edges, and 6 faces. (2) A planar graph consisting of 4 connected pieces must have Euler characteristic 5. (3) If a connected planar graph has 6 edges and splits the plane into 7 regions, then it must have 1 vertex. (a) (1) (2) (3) T T T (b) (1) (2) (3) T F T (c) (1) (2) (3) F T F (d) (1) (2) (3) F F F (e) (1) (2) (3) T F F (f) (1) (2) (3) T T F Solution Key: 2.10 Solution: 3.10 9 2 Solution key (1) (d) (2) (a) (3) (e) (4) (a) (5) (e) (6) (f) (7) (c) (8) (c) (9) see solution (10) (a) 10 3 Solutions Solution of problem 1.1: The altitude splits teh hypothenuse into two segments. If we denote the length of the shorter segment by x , then the length of the other segment will be 25 x . x 12 25 x a b If we denote the corresponding sides of the triangle by a and b respec tively we can apply the Pythagorean theorem to two small triangles and to the big triangle: x 2 + 12 2 = a 2 (25 x ) 2 + 12 2 = b 2 a 2 + b 2 = 25 2 Substituting the first two identities into the third one we get x 2 + (25 x ) 2 + 288 = 625 or x 2 + 625 50 x + x 2 + 288 = 625 . After cancelling 625 from both sides and dividing both sides by 2 we get the equation x 2 25 x + 144 = 0 . By the quadratic formula we have x = 25 ± √ 25 2 4 · 144 2 = 25 ± √ 49 2 = 25 ± 7 2 . 11 Thus x must be equal to either 9 or 16 , and since x was the shorter of the two segments we must have x = 9. Substituting this in the two identities involving a and b above we get a 2 = x 2 + 12 2 = 81 + 144 = 225 , b 2 = (25 x ) 2 + 12 2 = 256 + 144 = 400 . Therefore we have a = 15, b = 20, and a + b = 35. The correct answer is (d) . square Solution of problem 1.2: Denote the longer side of R 1 by x . Since R 1 is a golden rectangle we have that x/ 4 must be equal to the golden ratio ϕ , i.e. x = 4 ϕ. In particular R 1 has sides 4 and 4 ϕ , and R 2 has sides 4 ϕ and 4 + 4 ϕ . We can now compute the ratio of areas: area( R 2 ) area( R 1 ) = 4 ϕ · (4 + 4 ϕ ) 4 · 4 ϕ = 16 ϕ (1 + ϕ ) 16 ϕ = 1 + ϕ. The correct answer is (a) . square Solution of problem 1.3: Slicing off a vertex of cube creates a new trian gular face. Also, when we slice off a vertex all the original edges remain edges and we introduce three new edges  the sides of teh newly cre ated triangular face. Therefore, when we slice off the four top edges we introduce 4 · 3 = 12 new edges. Adding these to the original 12 edges the cube has we get a total of 24 edges for the new solid. The correctthe cube has we get a total of 24 edges for the new solid....
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 Spring '08
 schneps
 Math, Planar graph, Euler

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