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Unformatted text preview: Douglass Houghton Workshop, Section 1, Mon 11/28/11 Worksheet Omnibus 1. Shortest Network. So far we’ve looked at the case where the cities are at the corners of an isosceles triangle like the one at the right. We know: When you have a Y-shaped network like the one shown below, its total length is L θ ( x ) = x + 2 radicalBig x 2- 20 x cos( θ/ 2) + 100 . For x = 0, that’s just the V-shaped network with roads from A to B and B to C . When the vertex angle is 90 ◦ , then sometimes the Y beats the V . For instance, L 90 (3) ≈ 19 . 3 < 20 = L 90 (0) so placing the roundabout 3 miles south of B is better than not having a round- about at all. A B C 10 10 b b b θ b b b b A B C 10 10 x (a) What about θ = 150 ◦ ? Try to find an x which improves the V . (b) Put your calculator in degrees mode and plot L 90 ( x ) and L 150 ( x ), for x from 0 to 10. The shapes are about the same, but what’s the difference that explains the fact that you can improve some V-shaped networks and not others? (Remember-shaped networks and not others?...
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This note was uploaded on 01/28/2012 for the course MATH 146 taught by Professor Conger during the Winter '08 term at University of Michigan.
- Winter '08