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# w2o - Douglass Houghton Workshop Section 2 Tue Worksheet...

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Douglass Houghton Workshop, Section 2, Tue 11/29/11 Worksheet Out Among the Stars 1. Shortest Network. We found last week that some V newtorks can be improved to a Y , and some can’t. In particular, our big result was that: If A , B , and C form an isosceles triangle, then the V network is not minimizing if the vertex angle is less than 120 . Now we need to generalize to all possible placements of cities. (a) Prove that the network to the right is NOT minimizing. You don’t need to find the optimal network, just prove that this one can be improved. Hint: consider the portion of the network that is inside the circle. (b) What allowed that trick to work? Phrase your answer like this: “Any network which contains can be improved.” (c) Put it all together, and explain where the soap puts the roundabout. 2. Write the following sums in sigma ( ) notation. (a) 1 + 2 + 3 + 4 + · · · 10 (b) 1 + 2 + 3 + 4 + · · · + n (c) 3 + 5 + 7 + 9 + · · · + 21 (d) 4 + 9 + 16 + 25 + · · · + 100 (e) 2 . 3 + 2 . 8 + 3 . 3 + 3 . 8 + 4 . 3 + 4 . 8 + · · · + 10 . 3 (f) f ( a 1 ) + f ( a 2 ) + f ( a 3 ) + · · · + f ( a n ) 3. (From the Winter, 2011 Math 115 Final) The table below gives the expected growth

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w2o - Douglass Houghton Workshop Section 2 Tue Worksheet...

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