Douglass Houghton Workshop, Section 1, Mon 11/21/11
Worksheet Nigh on Thanksgiving
1. We’ve been working on the problem of finding the shortest road network between three
cities in the plane.
In the case we considered, the three cities were at the
corners of a 45
◦
45
◦
90
◦
triangle with legs 10 miles long.
The simplest idea is to just build roads along the legs;
that makes a network of length 20. But by constructing
a
Y
shaped network like the one at the right, we found
A
B
C
10
10
x
The length of the network is
x
+ 2
radicalBig
100

20
x
cos(45) +
x
2
.
We can improve from the simple 2road solution (
x
= 0, length = 20) by increasing
x
. For instance, when
x
= 5, the network has length 19.74.
(a) Consider the case where the triangle is still isosceles and
the legs still have length 10, but the angle at
B
is 70
◦
.
Write a formula for the length of the network.
(b) Can you find a value of
x
which beats the 2road solution
(
x
= 0, length = 20)?
A
B
C
x
(c) Now suppose the vertex angle is very obtuse—say
150
◦
. Find a formula for the length of the network.
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 Winter '08
 Conger
 The Network, road network, 2road solution

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