Advanced Algebra

Chapter 12 Answers Discrete Mathematics and Models

Graph theory has both theoretical and practical importance, as the following investigations show.

Part B, Paths and Circuits

1. Study the 13 Archimedean solids at this Web site to answer the following questions. (In the illustration, each solid is paired with its twin directly above or below it.)

a. How are Archimedean solids similar to Platonic solids? Different from Platonic solids?
[Like Platonic solids, Archimedean solids have faces that are regular polygons and have an equal number of edges meeting at each vertex. Unlike Platonic solids, Archimedean solids have faces that are not all congruent. Instead, they consist of two or even three different classes of regular polygons. The same polygons meet at every vertex.]

b. Show that Euler's Formula is valid for a truncated tetrahedron.
[F = 8, V = 12, E = 18; F + V E = 8 + 12 18 = 2]

c. Complete the table for the given polyhedra.

Polyhedron

Measures of the Angles Meeting at Each Vertex

S = Sum of Measures of Angles at Each Vertex

D + 360 - S

V = Number of Vertices

Cube
90°, 90°, 90°

270°

90°

8

Tetrahedron

Octahedron

Dodecahedron

Icosahedron

Truncated
Tetrahedron

Polyhedron

Measures of the Angles Meeting at Each Vertex

S = Sum of Measures of Angles at Each Vertex

D + 360 - S

V =
Number of Vertices

Cube
90°, 90°, 90°

270°

90°

8

Tetrahedron
60°, 60°, 60°

180°

180°

4

Octahedron
60°, 60°, 60°

240°

120°

6

Dodecahedron
108°, 108°, 108°

324°

36°

20

Icosahedron
60°, 60°, 60°

300°

60°

12

Truncated
Tetrahedron
60°, 20°, 120°

300°

60°

12

d. Find a relationship between D and V in the table from question c.
[DV = 720° . This relationshipo is known as Descartes' Formula for polyhedra.]

e. Use the relationship to find the number of vertices in a truncated icosahedron.
[S = 108 + 120= 348; D = 12 ÷ 12 = 60]

Part D, Making Connections

2. Use the Distance Web site to answer the following questions.

a. The graph shows eight Texas towns and the roads connecting them. Find the length of each road.

b. To visit all of her customers, a saleswoman completed an Euler path on the graph, beginning in Abilene. Where did the path end? Name the towns the saleswoman passed through and the order in which she passed through them.
[Coleman; Answers will vary; Abilene, Baird, Coleman, Winters, Abilene, Bronte, Winters, Ballinger, Bronte, San Angelo, Ballinger, Santa Anna, Coleman]

c. To visit warehouses in each town on the map, the saleswoman completed a Hamiltonian circuit on the graph, beginning in Abilene. Where did the path end? Name the towns the saleswoman passed through and the order in which she passed through them.
[She ended in Abilene. Answers will vary for the circuit followed, one possible answer is Abilene, Winters, Bronte, San Angelo, Ballinger, Santa Anna, Coleman, Baird, Abilene]

d. Find a minimal spanning tree for the graph. How long is it?
[Either of these two trees is minimal. Each has total length of 167 mi.]

Top

In the following activities, you will use recursion to explore changes in real estate prices and geometrical patterns.

Part A, Recursive Sequences

1. Use the Median Sale Price Web site to answer the following questions.

a. Find the table of median sale prices of existing single-family homes. Record the average single-family home selling prices for 1996 and 1997. [$118,000 and $123,600]

b. Find the percent change in average selling prices to the nearest hundredth.
[+4.75%]

c. Suppose you purchased a single-family home in 1996 at the average selling price for that year, that the value of the house changes at the rate you calculated in (1b), and that, each year, you make improvements that increase the value of your property by $3000. Give the value of the house in 1997, 1998, and 1999.
[$126,605; $135,618.74; $145,060.63]

d. Define a recursive sequence that gives Vn, the value of your property n years after purchase.
[V_n = 1.0475V_{n - 1} + 3000]

e. Give an explicit formula for the recursive sequence you have defined.
[V_n = 172,943.1(1.0475ⁿ) 63,157.895]

f. Find the value of your house after 10 years. By how much has the value increased over the purchase price?
[$211,912.31; $93,912.31]

g. How long will it take for the value of your house to double?
[172,943.1(1.0475ⁿ) 63,157.895 = 2 x $118,000
1.0475ⁿ = 1.7298
n = log 1.7298 / log 1.0475) = 11.8 years]

Part C, Making Connections

2. Find the grid for John Conway's game of Life.

a. Click on "the rules" to review the rules of the game. Then go to the Game of Life home page. Under "The Game," choose the initial generation called "Cross" and click on "initial generation".

b. How many generations does it take for the cross pattern to become stable? Sketch the stable pattern.
[7 generations;]

c. How many generations does it take for the "Diagonal Egg" to become stable? Why does the pattern have this property?
[one generation, i.e. the original pattern is stable; Each occupied square has 2 neighbors, so it always survives. No unoccupied cell, however, can become occupied because none has 3 neighbors.]

d. Describe the life cycle of the "Glider." Why do you think it is called the "Glider"?
[The pattern goes through a 4-generation cycle. At the end of each cycle the glider reforms in a new position 1 square to the right of and 1 square down from the previous position. Hence, the pattern "glides" slowly downward and to the right.]

e. Start with a blank grid. Design a pattern, with at least 5 occupied cells, that dies after one generation.
[Designs will vary. One possibility:

]

f. Start with a blank grid. Design a pattern, with at least 5 occupied cells, that is stable.
[Designs will vary. One possibility:

]