Skills Practice Skills Practice for Lesson 2.1

Transcription

1 Skills Practice Skills Practice for Lesson. Name Date Calories and Lunch Matrices and Problem Solving Vocabulary Write an example of each term given the description.. Write a 3 3 matrix.. Write an element of the matrix in Question. 3. Write a matrix equation using the matrix you wrote in Question. 4. Write an example of a row matrix. Carnegie Learning, Inc. 5. Write an example of a column matrix. Chapter l Skills Practice 349

2 Problem Set Determine the dimensions of each matrix.. A gift shop sells small, medium, and large fruit baskets. Each basket contains apples, oranges, pears, and bananas. The table shows the amounts of fruit in each size basket. Create a matrix and determine what a 3 represents. Baskets Apples Oranges Pears Bananas Small Medium 4 3 Large A a 3 represents the number of pears in a medium-sized basket, which is.. In addition to regular pizzas, a pizza shop sells three types of specialty pizzas for an additional cost. The types of specialty pizzas are supreme, veggie, and classic. The table shows the amounts of toppings needed to make each type of pizza, in addition to cheese and sauce. Create a matrix and determine what a 5 represents. Green Peppers (ounces) Pizzas Olives (ounces) Tomatoes (ounces) Pepperoni (ounces) Supreme Veggie Classic 5 6 Sausage (ounces) Carnegie Learning, Inc. 35 Chapter l Skills Practice

3 Name Date 3. Four friends are buying school supplies. They each buy pens, pencils, and notebooks. The table shows the number of each type of item that each person buys. Create a matrix and determine what a 4 represents. Person Pens Pencils Notebooks Lydia 6 6 Samuel 4 8 Marco Anna A company prints T-shirts for businesses with the business s name or logo. The T-shirt company sells long-sleeve and short-sleeve T-shirts. The table shows the number of each type of shirt ordered by two different companies. Create a matrix and determine what a represents. Carnegie Learning, Inc. Companies Long-Sleeve T-shirts Short-Sleeve T-shirts Smith Corporation 75 5 Prince Paper, Inc. 5 Chapter l Skills Practice 35

4 5. The table shows the approximate nutritional value in one slice of three different types of bread in a bakery. Create a matrix and determine what a 34 represents. Bread Carbohydrates (grams) Protein (grams) Fat (grams) Calories White 6 Multi-grain Raisin The table shows the approximate nutritional value in one serving of chicken soup and vegetable soup. Create a matrix and determine what a 3 represents. Carbohydrates Soup (grams) Protein (grams) Fat (grams) Calories Chicken Vegetable 4 96 Carnegie Learning, Inc. 35 Chapter l Skills Practice

5 Name Date Use matrices to solve each problem. You can use your graphing calculator to determine the multiplicative inverse. 7. A car salesperson receives a flat commission on pre-owned and new cars that she sells. The table shows the number of each type of cars that she sold in the past two months. Determine the commission that the salesperson receives for each type of car. Months Pre-owned New Commission (dollars) June July 3 A 5 3 A 7 7 A B , B The salesperson receives $5 commission for each pre-owned car that she sells and $5 commission for each new car that she sells. 8. A salesperson in an electronics store receives a flat commission for selling warranties on certain products. The table shows the number of each type of warranty the salesperson sold in the last two weeks. Determine the commission that the salesperson receives for selling each type of warranty. Carnegie Learning, Inc. Weeks -year Warranty 5-year Warranty Commission (dollars) Week Week Chapter l Skills Practice 353

6 9. Three friends are shopping in a thrift store. The table shows the number of shirts, jackets, and pairs of pants that each person buys. The last column represents the total cost of each person s purchase. Determine the cost of each item. Person Shirts Jackets Pairs of Pants Total Cost (dollars) Suzanne 5 7 Melinda 3 35 Leah 4. A rental car company rents out compact, mid-size, and luxury cars. The table shows the number of each type of car that the company rented out for a one-day rental in the past three months. The last column represents the rental company s revenue for one-day rentals in the past three months. Determine the cost for a one-day rental of each type of car. Months Compact Mid-Size Luxury Revenue (dollars) June July 6 3 August Carnegie Learning, Inc. 354 Chapter l Skills Practice

7 Name Date. A coffee shop sells small, medium, and large cups of coffee. The table shows the number of each size of coffee that the shop has sold in the past three days. The last column represents the amount of money the shop made from coffee sales. Determine the price of each size of coffee. Days Small Medium Large Amount Made (dollars) Monday Tuesday Wednesday The table shows the number of pounds of lunch meat that three customers purchased in a supermarket. The last column represents each customer s total cost for the lunch meat. Determine the price per pound for each type of lunch meat. Carnegie Learning, Inc. Customers Ham Roast Beef Salami Cost (dollars) Customer Customer Customer Chapter l Skills Practice 355

8 Carnegie Learning, Inc. 356 Chapter l Skills Practice

9 Skills Practice Skills Practice for Lesson. Name Date Regions and Vertices Systems of Linear Inequalities Vocabulary Explain how each set of terms are related.. system of linear inequalities and points of intersection. solution to system of linear inequalities and maximum value Carnegie Learning, Inc. Problem Set Graph the solution set for each system of linear inequalities. Label the point of intersection of the boundary lines. y 3x 4. x y y , ( ) 8 x y. x 3x y Chapter l Skills Practice 357

10 3x y 9 3. x y 4 6x y 4. x y 3 Graph the solution set for each system of linear inequalities. Label all points of intersection of the boundary lines. 5. y 4 x y y x 4 y y 4 6. x y x y 8 ( 8, 4) 6 4 y (7, 4) 4 6 (, 6) 8 x Carnegie Learning, Inc Chapter l Skills Practice

11 Name Date 7. y x y x x y 3 x y 8. y y x x y 3 y x y Carnegie Learning, Inc. 9. y y 5 x 3 x y 3x. y 3x y 4 x x 3y 3y x 8 Chapter l Skills Practice 359

12 Graph each system of linear inequalities and its solution region. Then graph the given lines on the same graph as the system. Describe the intersection of the given lines with the solution region to the system of inequalities.. y y x 3 x Given lines: ( 3, ) y 4 3 (, ) x y 3 x y 4 x y 9 The intersection of x y 3 and the solution region is a line segment with endpoints (3, ) and (, ) ( 3, ) (, ) x The intersection of x y 4 and the solution region is the point (, ). There is no intersection for x y 9 and the solution region.. y 3 y x x 3 Given lines: x y x y x y 4 Carnegie Learning, Inc. 36 Chapter l Skills Practice

13 Name Date 3. y x y x x y 4 Given lines: y y 3 y 5 4. y y x x x y 6 Carnegie Learning, Inc. Given lines: x 3y 3 x 3y 9 x 3y Chapter l Skills Practice 36

14 5. x y 4 y 4x 3x y Given lines: x y 4 x y 5 x y 6 6. x 3 y 4 x y 4 x y 8 Given lines: x y x y x y 5 Carnegie Learning, Inc. 36 Chapter l Skills Practice

15 Name Date Each graph shows the solution region of a system of linear inequalities. Graph the given lines on the same graph as the system of linear inequalities. Then determine the maximum value of the indicated expression. y 7. Given lines: x y 5 x y x y x y 8 4 (, ) (, 3) 3 4 x What is the maximum value of x y in the solution region? The maximum value of x y is at the point (, 3) and is 5. (, 4) 3 4 (, 3) 8. 8 y 6 (, 4) 4 (3, 4) Carnegie Learning, Inc Given lines: x y x y (, ) (7, ) x x y 4 x y 7 What is the maximum value of x y in the solution region? Chapter l Skills Practice 363

16 9. Given lines: x y 8 6 y x y x y 4 4 (,.8) (6,.8) x y x 4 (, 8 3 ) 6 8. y 8 ( 8, 5) 6 4 (4, 5) 8 ( 3, ) 6 4 (, ) x Given lines: x 3y 6 x 3y 3 x 3y 9 x 3y 3 Carnegie Learning, Inc. 364 Chapter l Skills Practice

17 Name Date. Given lines: x 4y y 8 x 4y 6 x 4y 8 x 4y (3, 3) 4 6 (3, ) 8 x (, ) 4 (, 6) 6 8. y (, ) (6.5, ) x ( 6, 3) 4 (.5, 3) 6 Carnegie Learning, Inc. Given lines: 5x y 5x y 4 5x y 5 5x y Chapter l Skills Practice 365

18 Carnegie Learning, Inc. 366 Chapter l Skills Practice

19 Skills Practice Skills Practice for Lesson.3 Name Date Constraints and Solutions Linear Programming Vocabulary Define each term in your own words.. linear programming. constraints Problem Set Graph the solution set for each system of linear inequalities. Label all points of intersection of the boundary lines. Carnegie Learning, Inc.. y 4 x 3 y x x y ( 3, 4) y 4 3 (, 4). y x 3 y x x 3y (, ) 3 4 x ( 3, 3) 3 4 Chapter l Skills Practice 367

20 3. y x x y 4 3x y 5 4. x 3 x y x 4x y 5. x y 5 x 3y y 8x y 6. x y x y y y Carnegie Learning, Inc. 368 Chapter l Skills Practice

21 Name Date Each graph shows the solution region of a system of linear inequalities. Evaluate the given expression at each of the vertices of the solution region and identify the point at which the expression has a maximum value. 7. y 8. y (3, 7) (, 5) (5, ) (9, 5) (, 3) 3 (, 5) (, ) (6, ) x Given expression: x 3y Given expression: 8x 5y x 3y () 3(5) 7 x 3y (3) 3(7) 7 x 3y (9) 3(5) 33 x 3y (5) 3() 3 Carnegie Learning, Inc. The maximum value of x 3y is at the point (9, 5) and is 33. Chapter l Skills Practice 369

22 9. y. y (, 7) (, 5) 4 3 (3, 4) (, 4) (3,.5) (3, 3.5) x x Given expression: x y Given expression: 3x y Carnegie Learning, Inc. 37 Chapter l Skills Practice

23 Name Date. y. y (, 6) (, 4) (3, 6) (4, 4) x (, 5) (3, 5) (, ) x Given expression: x 3y Given expression: 3x 4y Carnegie Learning, Inc. Chapter l Skills Practice 37

24 Write a system of inequalities that represents each situation. Be sure to define the variables. 3. A company manufactures two types of computers: an 8-megahertz (MHz) computer and a faster -gigahertz (GHz) computer. The 8 MHz computer requires 3 person-hours to produce. The GHz computer requires 5 person-hours to produce. The company has a total of 8 person-hours available each day for the production of the computers. There is enough space in the shipping warehouse for the company to produce a maximum of computers per day. Let x represent the number of 8 MHz computers produced. Let y represent the number of GHz computers produced. 3x 5y 8 x y x y 4. The Cedar Brook Paper Company produces two different types of photo paper: glossy and matte. The most the company can produce in one day is 5 reams of photo paper. It costs $35 to produce a ream of glossy paper and $3 to produce a ream of matte. The most the company wants to spend in one day on production is $7,5. 5. A school is having a bake sale and needs at least 5 dozen cookies to sell. The cookies will come in a small and large variety. The small cookies will be sold for $.5 per dozen and the large cookies will be sold for $ per dozen. They need to make at least $5 to be able to buy T-shirts for the school dance. Carnegie Learning, Inc. 37 Chapter l Skills Practice

25 Name Date 6. A farmer has acres to plant corn and soybeans. It costs him $5 per acre to plant corn and $7 per acre to plant soybeans. The farmer has $8, budgeted for this season. 7. A small calculator company produces scientific and graphing calculators. The company can produce at most 4 calculators each day. It costs the company $3 to make each scientific calculator, and they make a profit of $ on each. It costs the company $ to make each graphing calculator, and they make a $ profit on each. Each day of production, they need to spend less than $5 on production and make a profit of at least $. Carnegie Learning, Inc. 8. An office manager needs to purchase file cabinets. Small file cabinets cost $75 and take up 6 square feet of floor space. Large file cabinets cost $5 and take up 8 square feet of floor space. He has been given $ to buy file cabinets and wishes to take up no more than 5 square feet of floor space. Chapter l Skills Practice 373

26 Use the given information to answer each question. Show all your work. 9. A small pet food company produces and sells all natural cat food and dog food. It costs the company $.35 to produce one pound of cat food and $. to produce one pound of dog food. The daily production costs for producing the cat and dog food cannot exceed $5. The company has the resources to produce a maximum of pounds of pet food each day. The pet food company makes a profit of $. per pound for the cat food and $.5 per pound for the dog food. How many pounds of cat food and how many pounds of dog food should the company produce to maximize their profit, and what is the maximum profit? y (, ) 5 ( 666, ) 5 (48.57, ) x Let x represent the number of pounds of cat food produced. Let y represent the number of pounds of dog food produced..35x.y 5 x y x y Profit: x.5y x.5y ().5() x.5y ().5() 3 x.5y ( ).5 ( ) x.5y ( ()) The pet food company should produce 666 pounds of cat food and pounds of dog food, for a maximum profit of $ Carnegie Learning, Inc. 374 Chapter l Skills Practice

27 Name Date. A packaging company produces and sells small and large boxes. The company can produce no more than 5 boxes each day. It costs the company $. to produce each small box and $.4 to produce each large box. The daily production costs for producing the boxes cannot exceed $5. The profit for the small boxes is $5 per box and the profit for the large boxes is $7.5 per box. How many small boxes and large boxes should the company produce to maximize their profit, and what is the maximum profit? Carnegie Learning, Inc. Chapter l Skills Practice 375

28 . A company produces two different types of specialty nails by hand. It costs the company $4 to produce each pound of small nails and takes man hours to complete. It costs the company $5 to produce each pound of large nails and takes 4 man hours to complete. The company has, at most, man hours available each day and wants to spend less than $3 each day on production. The profit on each pound of small nails is $ and the profit on each pound of large nails is $5. How many pounds of small nails and large nails should the company produce to maximize their profit, and what is the maximum profit? Carnegie Learning, Inc. 376 Chapter l Skills Practice

29 Name Date. A painting company is placing a bid for a job. They can take up to 5 days to paint at most 3 offices in an office building. Team A of the company can paint 3 rooms per day. Team B can paint 5 rooms per day. The office manager has determined that using Team A earns a profit of $85 per day and Team B earns a profit of $7 per day. How many days should the company use Team A and how many days should they use Team B to maximize their profit, and what is the maximum profit? Carnegie Learning, Inc. Chapter l Skills Practice 377

30 3. Liza knits scarves and hats and sells them online. It takes her 6 hours to knit a scarf and 8 hours to knit a hat. Liza can spend no more than hours each month knitting, because she has another job. She can afford enough yarn to knit no more than 4 items each month. Liza makes a profit of $8 from each scarf she sells and $ for each hat she sells. How many scarves and how many hats should Liza knit each month to maximize her profit, and what is her maximum profit? Carnegie Learning, Inc. 378 Chapter l Skills Practice

31 Name Date 4. Tom works at a landscaping business and a supermarket stocking shelves after school and on weekends. Each week, he can work no more than 8 hours. He must work at least 8 hours per week at the landscaping business and 5 hours per week at the supermarket stocking shelves. He earns $5.5 per hour at the landscaping business and $7 per hour at the supermarket. How many hours should Tom work at the landscaping business and at the supermarket to earn the most money? Carnegie Learning, Inc. Chapter l Skills Practice 379

32 Carnegie Learning, Inc. 38 Chapter l Skills Practice

33 Skills Practice Skills Practice for Lesson.4 Name Date Spies a Lot! Introduction to Directed Graphs Vocabulary Use the diagram to give an example of each term.. arc. directed edges W a 5 a Y a X a 3 a 4 3. vertices Z 4. nodes 5. origin of an arc Carnegie Learning, Inc. 6. initial node of an arc 7. destination of an arc 8. terminal node of an arc Chapter l Skills Practice 38

34 9. parallel arcs. loop Write the term that best completes each statement. adjacency matrix directed graph digraph length of a path simple diagraph successor. The diagram above shows a(n) or a(n).. In the diagram above, P( X, a, Y, a 4, Z ) represents the that starts at X, passes through Y, and ends at Z. 3. A diagraph without parallel arcs or loops is called a(n). 4. In the directed edge a (P, Q), Q is the of P. 5. The is the number of arcs in a path. 6. A(n) is a way to represent the arcs of a diagraph. If an arc exists, a is used. If the arc does not exist, a is used. Problem Set Given each scenario, draw arrows to show how information is passed between the assets, handlers, and supervisors.. Consider a spy network consisting of two supervisors, S and S, two handlers, H and H, and four assets, A, A, A 3, and A 4. Assets A and A report to handler H. Assets A 3 and A 4 report to handler H. Asset A also reports directly to supervisor S, as does handler H. Asset A 4 also reports directly to supervisor S, as does handler H. The supervisors report any information either receives to each other. Carnegie Learning, Inc. S A 4 H S A 3 A H A 38 Chapter l Skills Practice

35 Name Date. Consider a spy network consisting of two supervisors, S and S, two handlers, H and H, and six assets, A, A, A 3, A 4, A 5, and A 6. Assets A, A and A 3 report to handler H. Assets A 4, A 5, and A 6 report to handler H. Asset A 3 also reports directly to supervisor S, as does handler H. Assets A 4, and A 5 also report directly to supervisor S, as does handler H. The supervisors report any information either receives to each other. S H A A S A 3 H A 4 A 5 A 6 3. Consider a spy network consisting of one supervisor, S, three handlers, H, H, and H 3, and four assets, A, A, A 3, and A 4. Assets A and A report to handler H. Asset A 3 reports to handler H and asset A 4 reports to handler H 3. Handlers H and H 3 report to handler H. Handler H also directly reports to the supervisor S. Handlers H and H 3 collaborate any information either receives to each other before making a report. Carnegie Learning, Inc. S H H H 3 A A A 3 A 4 Chapter l Skills Practice 383

36 4. Consider a spy network consisting of three supervisors, S, S, and S 3, two handlers, H and H, and five assets, A, A, A 3, A 4, and A 5. Assets A, A, and A 3 report to handler H. Assets A 4 and A 5 report to handler H. The handlers H and H report to all three supervisors. The handlers report any information either receives to each other before reporting to a supervisor. S S 3 A5 S H A H A A 3 A4 Identify the origin (or initial node), destination (or terminal node), and successor of the directed edge of each digraph described. 5. Suppose that c (B, T ) is a directed edge of digraph D. Identify the origin of c, the destination of c, and the successor of B. B is the origin of c. T is the destination of c. T is the successor of B. 6. Suppose that e (F, M ) is a directed edge of digraph K. Identify the origin of e, the destination of e, and the successor of F. Carnegie Learning, Inc. 7. Suppose that k (R, T 3 ) is a directed edge of digraph L. Identify the origin of k, the destination of k, and the successor of R. 384 Chapter l Skills Practice

37 Name Date 8. Suppose that e 3 (M, R ) is a directed edge of digraph Q. Identify the origin of e 3, the destination of e 3, and the successor of M. For each digraph D shown, define D(V ), D(E ), and each edge as an ordered pair. 9. W a X a 5 Y a a 3 a 4 Z D(V ) W, X, Y, Z D(E ) a, a, a 3, a 4, a 5 Edges defined as ordered pairs: a (X, W), a (X, Y), a 3 (Y, Z), a 4 (Y, Z ), a 5 (W, W). a 4 A a 5 Carnegie Learning, Inc. B a 3 a D E D(V ) a C a 6 D(E ) Edges defined as ordered pairs: Chapter l Skills Practice 385

38 . e 4 S R e 5 e 3 e e 6 T U e. D(V ) D(E ) Edges defined as ordered pairs: a M b N e d O c f D(V ) D(E ) Edges defined as ordered pairs: 3. f 8 D H f f 4 f 6 f 5 f 7 G D(V ) E f3 F f Carnegie Learning, Inc. D(E ) Edges defined as ordered pairs: 386 Chapter l Skills Practice

39 Name Date 4. C B a 4 a a 5 A a a 3 D D(V ) D(E ) Edges defined as ordered pairs: Sketch each digraph given its description. 5. Digraph J is defined as: J(V ) P, Q, R, S, T J(E ) a, b, c, d, e, f, g a (P, Q), b (P, Q), c (P, S), d (S, T), e (T, S), f (Q, R), g (S, R) P c g S a b R d e Q f T Carnegie Learning, Inc. 6. Digraph P is defined as: P(V ) L, M, N, O P(E ) a, a, a 3, a 4, a 5 a (L, M), a (M, N), a 3 (N, M), a 4 (L, O), a 5 (O, L) Chapter l Skills Practice 387

40 7. Digraph R is defined as: R(V ) X, Y, Z R(E ) h, i, j, k, l h (X, Y ), i (Y, Y ), j (X, Z ), k (Z, X ), l (Z, Z ) 8. Digraph P is defined as: P(V ) R, S, T, U P(E ) h, h, h 3, h 4 h (R, S), h (T, S), h 3 (U, S ), h 4 (S, U ) 9. Digraph K is defined as: K(V ) F, G, H, I, J K(E ) r, s, t, u, v, w, x r (F, G), s (G, H ), t (H, H ), u (I, J ), v (J, I ), w (F, I ), x (F, J ) Carnegie Learning, Inc. 388 Chapter l Skills Practice

41 Name Date. Digraph Z is defined as: Z(V ) A, B, C, D, E Z(E ) l, m, n, p, q, r, s, t l (A, B), m (A, C), n (B, C), p (E, C), q (B, E ), r (E, D), s (D, C), t (D, A) Determine the length of each path given the description of the digraph.. Digraph L is defined as: L(V ) F, G, H, J L(E ) a, b, c, d, e a (F, J ), b (G, F ), c (F, G), d (H, F ), e (G, H ) What is the length of path P, where P (F, c, G, e, H, d, F )? The length of path P is 3.. Digraph M is defined as: M(V ) A, B, C, D Carnegie Learning, Inc. M(E ) g, h, i, j, k, l g (B, A), h (D, B), i (C, D), j (A, C), k (B, C), l (A, D) What is the length of path P, where P (A, j, C, l, D, h, B)? 3. Diagraph T is defined as T(V ) M, N, O, P T(E ) a, a, a 3, a 4, a 5 a (M, N ), a (O, N ), a 3 (N, P), a 4 (M, P), a 5 (P, O) What is the length of path P, where P (M, a, N, a 3, P, a 5, O)? Chapter l Skills Practice 389

42 4. Digraph A is defined as A(V ) Q, R, S, T, U A(E ) x, x, x 3, x 4, x 5, x 6 x (Q, R), x (R, S), x 3 (S, T ), x 4 (T, U ), x 5 (U, Q), x 6 (U, S) What is the length of path P, where P (Q, x, R, x, S, x 3, T, x 4, U)? Write the adjacency matrix A for each digraph. 5. J a K 6. M x N x 3 x 4 a x a 3 O L A J K L J K L 7. Y a 5 a a X a 4 W a 3 8. R c a Q S d b Carnegie Learning, Inc. 39 Chapter l Skills Practice

43 Name Date 9. M a b N 3. x Q c d e P B x x 3 C x 4 x 6 D x 5 E The given matrix A is an adjacency matrix for a digraph. Use the given information to answer each question. 3. Given the matrices below, how many paths of length two are there from B to C? A B C D B C D A A3 There is path of length two from B to C. A Given the matrices below, how paths of length two are there from R to S? Carnegie Learning, Inc. A R S T R S T A 3 3 A A Given the matrices below, how many paths of length three are there from R to R? A P Q R P Q R A A3 3 A4 3 4 Chapter l Skills Practice 39

44 34. Given the matrices below, how many paths of length three are there from L to M? A L M N L M N A A3 3 A Given the matrices below, how many paths of length four are there from Y to X? A W X Y W X Y A A3 3 A4 36. Given the matrices below, how many paths of length four are there from D to E? 3 4 A C D E C D E A A3 3 4 A Each matrix is an adjacency matrix for a digraph. Use the adjacency matrix to list the number of paths of length two between each pair of vertices for each digraph. 37. A X Y Z X Y Z A There are paths of length between X and Y, Y and X, Z and X, Z and Y, and Z and Z. 38. A There is path of length between X and X, X and Z, Y and Y, and Y and Z. B C D B C D Carnegie Learning, Inc. 39 Chapter l Skills Practice

45 Name Date F G H J F 39. A G H J T U V W T 4. A U V W Carnegie Learning, Inc. Each matrix is an adjacency matrix for a digraph. Use the adjacency matrix to list the number of paths of length three between each pair of vertices for each digraph. 4. A P Q R P Q R A3 There are paths of length 3 between Q and P. There is path of length 3 between P and P, Q and Q, and R and R. There are paths of length 3 between P and Q, P and R, Q and R, R and P, and R and Q. Chapter l Skills Practice 393

46 4. A R S T R S T 43. A D E F G D E F G G 44. A H I J G H I J Carnegie Learning, Inc. 394 Chapter l Skills Practice

47 Skills Practice Skills Practice for Lesson.5 Name Date Electricity and Water! Directed Graphs and Problem Solving Vocabulary Explain how each set of terms are related.. directed graphs and digraphs. digraph and adjacency matrix 3. matrix and element of a matrix Problem Set Draw a digraph for each adjacency matrix A. Carnegie Learning, Inc.. A X Y Z X Y Z X P Q R P. A Q R Y Z Chapter l Skills Practice 395

48 396 Chapter l Skills Practice Carnegie Learning, Inc. L M N P B C D E 3. A L M N P 4. A B C D E R S T U V F G H I J 5. A R S T U V 6. A F G H I J

49 Name Date Define an adjacency matrix A for each digraph. 7. R S 8. Z X Y T A R S T R S T 9. R S. J K T U L M Carnegie Learning, Inc. Chapter l Skills Practice 397

50 . T. B F P Q S C E R D Determine the number of indicated routes that exist. 3. A weather station has four different satellites, J, K, L, and M, which transmit and receive signals. The adjacency matrix A below represents the satellites that transmit and receive signals. How many two-step routes exist from satellite L to satellite K? List the route(s). J K L M J K A L M A There is two-step route from L to K: (LMK). Carnegie Learning, Inc. 398 Chapter l Skills Practice

51 Name Date 4. A cable company has four different signal hubs, P, Q, R, and S, which transmit and receive signals. The adjacency matrix A below represents the hubs that transmit and receive signals. How many two-step routes exist from hub P to hub Q? List the route(s). A P Q R S P Q R S 5. A navigation system has four different satellites, R, S, T, and U, which transmit and receive signals. The adjacency matrix A below represents the satellites that transmit and receive signals. How many two-step routes exist from satellite S to satellite T? List the route(s). A R S T U R S T U Carnegie Learning, Inc. Chapter l Skills Practice 399

52 6. A phone company has four different stations, B, C, D, and E, which transmit and receive signals. The adjacency matrix A below represents the stations that transmit and receive signals. How many three-step routes exist from station C to station E? List the route(s). A B C D E B C D E 7. A company has four different stores, G, H, I, and J, which send and receive interoffice shipments to each other. The adjacency matrix A below represents the stores that send and receive interoffice shipments. How many three-step routes exist from store H to store G? List the route(s). A G H I J G H I J Carnegie Learning, Inc. 4 Chapter l Skills Practice

53 Name Date 8. An electric company has five different grids, M, N, P, Q, and R, which send and receive signals to each other. The adjacency matrix A below represents the grids that send and receive signals. How many two-step routes exist from grid Q to grid M? List the route(s). A M N P Q R M N P Q R Identify the indicated element of each sum matrix and determine what it represents in the problem situation. 9. A radio network has four different transmission towers, P, Q, R, and S. The adjacency matrix A below represents the towers that transmit radio waves. Identify the element a 34 of the sum matrix A A and determine what this element represents in the problem situation. Carnegie Learning, Inc. A P Q R S P Q R S A A The element a 34 of the sum matrix is. It represents the number of paths from tower R to tower S of length or less. The path is (RS) and its length is. Chapter l Skills Practice 4

54 . A satellite network has four different satellites, B, C, D, and E. The adjacency matrix A below represents the satellites that send and receive signals. Identify the element a of the sum matrix A A and determine what this element represents in the problem situation. A B B C D E C D E. A phone company has four different main stations, X, R, T, and V. The adjacency matrix A below represents the stations that send and receive signals. Identify the element a 43 of the sum matrix A A and determine what this element represents in the problem situation. A X X R T V R T V Carnegie Learning, Inc. 4 Chapter l Skills Practice

55 Name Date. A machine has four different sensors, P, Q, R, and S. The adjacency matrix A below represents the sensors that send and receive signals. Identify the element a 3 of the sum matrix A A A 3 and determine what this element represents in the problem situation. A P Q R S P Q R S 3. A company has four different stations, M, N, O, and P. The adjacency matrix A below represents the sensors that send and receive signals. Identify the element a 4 of the sum matrix A A A 3 and determine what this element represents in the problem situation. Carnegie Learning, Inc. A M N O P M N O P Chapter l Skills Practice 43

56 4. A delivery company has five different delivery stations, E, F, G, H, and I. The adjacency matrix A below represents the stations that send and receive packages. Identify the element a 5 of the sum matrix A A and determine what this element represents in the problem situation. A E F G H I E F G H I Carnegie Learning, Inc. 44 Chapter l Skills Practice

57 Chapter l Skills Practice 45 Carnegie Learning, Inc. Name Date Use the given information to answer each question. 5. The digraph below represents the five different countries, B, C, D, E, and F, and their importing and exporting patterns. Each arrow represents goods that are imported and exported from one country to another. Did goods from every country reach every other country in two or fewer transactions? If not, list them. C B D E F B C D E F adjacency marix A B C D E F A A 3 3 The goods from country F did not reach country E in two or fewer transactions.

58 6. The digraph below represents the flow routes of water pipes and five valves, G, H, I, J, and K. Each arrow represents water flowing from one valve to another. Did water from every valve reach every other valve in two or fewer transactions? If not, list them. G H I J K Carnegie Learning, Inc. 46 Chapter l Skills Practice

59 Name Date 7. The digraph below represents the alarm sensors in a house, U, V, W, X, and Y. Each arrow represents signals moving from one sensor to another. Did signals from every sensor reach every other sensor in two or fewer transactions? If not, list them. U V W X Y Carnegie Learning, Inc. Chapter l Skills Practice 47

60 8. The digraph below represents the computers in a network, E, F, G, and H. Each arrow represents signals moving from one computer to another. Did signals from every computer reach every other computer in three or fewer transactions? If not, list them. E F G H Carnegie Learning, Inc. 48 Chapter l Skills Practice

61 Name Date 9. The digraph below represents the towers in a radio network, N, O, P, and Q. Each arrow represents signals moving from one tower to another. Did signals from every tower reach every other tower in three or fewer transactions? If not, list them. N O Q P Carnegie Learning, Inc. Chapter l Skills Practice 49

62 3. The digraph below represents the trade routes of five ancient cities, W, R, P, and T. Each arrow represents goods moving from one city to another. Did goods from every city reach every other city in three or fewer transactions? If not, list them. W R P T Carnegie Learning, Inc. 4 Chapter l Skills Practice