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# 3 if a planar graph has 2 connected pieces 12 edges

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(3) If a planar graph has 2 connected pieces, 12 edges, and 7 vertices, then it splits the plane into 8 regions. (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) F F T (f) (1) (2) (3) F T T Solution Key: 2.8 Solution: 3.8 Question 9: The fish population in a small pond is modeled by the Verhulst formula P n +1 = P n + cP n (1 - P n ), where P n stands for the fish popula- tion density after n years. The pond can sustain 300 fish and has initial fish population of 1200 fish. If after two years the fish population is 600 fish, how big will the fish population be after three years? (a) 200 fish (b) 300 fish (c) 400 fish (d) 500 fish (e) 600 fish (f) 700 fish Solution Key: 2.9 Solution: 3.9 8

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Question 10: You roll three dice. What is the probability that you will get three numbers that add up to 6? Solution Key: 2.10 Solution: 3.10 Question 11: Suppose 4 students out of 100 cheated during the last year, and you used the two-coin method to survey those 100 students. How many students would you expect to answer “yes”? Solution Key: 2.11 Solution: 3.11 Question 12: It is known that 8 , 000 students go to school in a college town. A local supermarket surveys 720 customers and determines that 180 of them are college students. Based on this information, estimate the non-student population of the town. Solution Key: 2.12 Solution: 3.12 9
Question 13: John deposited \$ 1 , 600 in savings account. After two years John had \$ 1 , 681 in the account. What was the annual percentage interest paid by this account? (a) 2 % (b) 2 . 7 % (c) 1 . 4 % (d) 1 . 8 % (e) 2 . 5 % (f) 3 . 1 % Solution Key: 2.13 Solution: 3.13 Question 14: A class of 110 people is voting for a new class president. The voters are choosing among four candidates by using the ranked voting system. Each voter ranks all four candidates: 1, 2, 3, 4 with 1 being the highest ranking. The candidates are then ordered by a total numerical score - the candidate with a lowest numerical score has the highest ranking in the outcome, the candidate with the next numerical score has the second highest ranking, etc.. Which of the following properties of a fair voting system are not sat- isfied for this voting system? Circle all that apply. Explain your rea- soning. Solution Key: 2.14 Solution: 3.14 10

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2 Solution key (1) (f) (2) (a) (3) (e) (4) (b) (5) (f) (6) (b) (7) (d) (8) (b) (9) (d) (10) (e) (11) (f) (12) (a) (13) (d) (14) (a) 11
3 Solutions Solution of problem 1.1: Let A N be the number of all natural numbers that can be written with the digits 1 and 2 only and such that the sum of all digits is equal to N . We can compute A N recursively: If N = 1, then there is only one such number: the number 1. Thus A 1 = 1. If N = 2, then there are two such numbers: 11 and 2. Thus A 2 = 2.

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• Fall '08
• schneps
• Natural Numbers, Rational number, Irrational number, Verhulst

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