A C E. Answers Investigation 4. Applications. Shape Number Number of Toothpicks

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1 Applications 1. a. (See Figure 1.) b. possible equation: T = 3s, where s is the shape number and T is the number of toothpicks c. There are many equations: for example, T = s + s + s or T = s + 2s would also model the relationship. 2. a. (See Figure 2.) b. possible equation: T = s + s + 2 c. Ahna s pattern does not grow as much because only two of the sides are growing from one shape to the next, while Scott s has all three sides growing for each shape. 3. a. (See Figure 3.) b. possible equation: T = s s + 1 c. Lloyd s pattern and Ahna s pattern use the same number of toothpicks. You could imagine moving the top toothpick in Lloyd s pattern down to the bottom and making the U shape in Ahna s pattern. 4. a. (See Figure 4.) b. possible equation: c = 2n + 3, c = number of cubes c. Answers may vary. A student might say yes because you could build the frame using five cubes, which would be consistent with the table in part (a). Given the context, however, the swing would not get off the ground if it was 1 cube high. Figure 1 Shape Number Number of Toothpicks Figure 2 Shape Number Number of Toothpicks Figure 3 Shape Number Number of Toothpicks Figure 4 Height (squares) Number of Cubes Variables and Patterns 1

2 d. Possible answer: Like the frames shown, a frame 50 cubes tall has 3 cubes in the middle. It is much taller than the frames shown, since it has 50 cubes on each side. The total number of cubes is = a. (See Figure 5.) b. All of the expressions are equivalent. Given any number of squares tall, each expression gives the same number of cubes. c. possible new expression: 2T a. (See Figure 6.) b. (See Figure 7.) c. (See Figure 8.) d. (See Figure 9.) Martha s, Chad s, and Jeremiah s equations are equivalent. 7. a. B = n b. L = 49n c. H = 125n d. V = a. C = ( n) + 49n + 125n + 95 b. C = 199n Figure 5 Number of Squares Tall Number of Cubes (Mitch) Number of Cubes (Lewis) Number of Cubes (Corky) Figure 6 Number of Squares Number of Pieces (Martha) Figure 7 Number of Squares Number of Pieces (Chad) Figure 8 Number of Squares Number of Pieces (Jeremiah) Figure 9 Number of Squares Number of Pieces (Lara) Variables and Patterns 2

3 c. Three possible explanations: Explanation 1: Tables of sample cost values are identical, as are graphs of the two relationships. Explanation 2: Each person s separate costs (25, 49, and 125) are equal to a total of 199 per person, and the 100 and 95 are fixed independent of number of customers. Explanation 3: Use the Distributive Property to find 25n + 49n = ( )n = 74n. Again use the Distributive Property to find 74n + 125n = ( )n = 199n. Then the expression n + 95 is equal to 199n because of the Commutative Property, and that expression simplifies to 199n a. T = n b. H = n 10. a. C = ( n) + ( n) b. C = 17.5n c. Two possible explanations: Tables and graphs of sample (n, C) values are identical. The Commutative Property of Addition tells you ( n) + ( n) = n + 10n, and the Distributive Property tells you n + 10n = n. The last expression is the same as 17.5n a. The terms in the expression 350n 30n ( n) could be defined as 350n, 30n, 350, and ( n). Some might consider 50 and 10n as separate terms. b. The numbers 350, 30, 350, and 10 are coefficients. Some might write 30 instead of 30 and mention 1 or 1 as a coefficient for the term ( n). c. The word term is used to describe parts of the algebraic expression separated by an addition or subtraction operation sign. When a term is a product of a number and a variable, the word coefficient is used to describe the number. 12. The equation 500 = 2(150) + 2(x) has solution x = Superior Bus: C = 2.95m; Coast Transport: C = m 14. The solution for 590 = 2.95m is m = 200. Check: 2.95(200) = 590. The easiest 15. The solution for 600 = m is m = 150. Check: (150) = 600. Possible explanations: Inspect a table or graph of the equation R = m, looking for values of the independent variable m that produce a value of 600 for the dependent variable R. Use inverse operations subtract 300 from 600 and then divide by a. Equation is 2.95m = m. The table shows charges are equal for about 300 miles, so that is the first guess. Exact distance is 316. b. The charge by Superior Bus will be less than that by Coast Transport for travel of less than about 300 miles (precisely 315 miles). c. The charge by Coast Transport will be less than that by Superior Bus for travel of more than 300 miles (precisely 316 miles). 17. a. 3,700 = 190n 1,050 when n = 25. The bike tour will make a profit of about $3,700 when it has 25 customers. b. 550 = 190n 1,050 when n 8.4. The bike tour will make a profit of about $2,750 when it has 8 or 9 customers. 18. a. Values in the table suggest that the solution to 1,230 = 190 1,050 lies about midway between 10 and 15. b. The exact solution is 12. Check: 190(12) 1,050 = 1,230. c. i. The solution to 2,560 = 190n 1,050 is 19. With 19 customers, the tour will make a profit of $2,560. ii. The solution to 5,030 = 190n 1,050 is 32. With 32 customers, the tour will make a profit of $5,030. Variables and Patterns 3

4 19. a. The graph shows the solution is about 16. Check: 190(16) 1,050 = 1,990. b. i. The graph of y = 5x + 10 shows that the equation 45 = 5x + 10 has a solution of about 7. Substituting 7 for x gives 45 = 5(7) + 10, which checks exactly. ii. The graph of y = x shows that the equation 60 = x has a solution of about 16. Substituting 16 for x gives 60 = (16), which checks exactly Equation is 2,180 = 190n 1,050. Solution is n = The arithmetic operations 2, ,050 and 3, give the solution. Connections 21. a b = 7a x y = x m n = 2m r s = r Variables and Patterns 4

5 25. a. (See Figure 10.) b. s = 4c Students should create tables or graphs to justify whether the expressions are equivalent. a. equivalent b. Not equivalent; any value besides m = 1 will result in different values for the expressions. c. Not equivalent; any value besides p = 7 will result in different values for the expressions. d. Not equivalent; the two expressions are never equal (they always differ by 2). e. equivalent 27. possible answer: 5n = n + 4n 28. possible answer: 2n + 2 = 2(n + 1) 29. possible answer: 4n 4 = 4(n 1) 30. possible answer: 3n + 2n + n = 6n 31. 4(3 + 9) = 4(3) + 4(9) 32. 5(n + 7) = 5n + 5(7) = 7(11) + 7(3) = 7(11 + 3) Students should consider what common factors there are for 21 and 77. If answers are not limited to whole numbers, there are many possibilities n + 40 = 8n + 8(5) = 8(n + 5) 35. 4(7) + 4(5) = 4(7 + 5) 36. 3n + 12n = n(3 + 12) 37. 3(2) + 3(4) + 3(2) = 3( ) 38. n(n) + 5n = n(n + 5) = n 2 + 5n 39. a. 10(13 + 7) = 10(20) = 10(13) + 10(7) b. ( )20 = 10(20) = 7.5(20) + 2.5(20) < = > = > Figure 10 Relationship of Cubes to Squares in a Tower Cubes in the Tower Squares in the Tower Variables and Patterns 5

6 > > < three terms, coefficients 4, 5, four terms, coefficients 6, 4, 1, four terms, coefficients 2, 3, 2, three terms, coefficients 5, 1, a. c = p + t b. t = 0.08p c. c = p p d. c = 1.08p 53. x = 19 when x = 5.5. Solve by calculating Check: = = x 7 when x = 30. Solve by calculating Check: 23 = x = 405 when x = 9. Solve by calculating 405. Check: 45(9) = x 11 = 37 when x = 6. Solve by calculating ( ) = 8. Check: 8(6) 11 = a = 15 is equivalent to 8 = 15 7 and 7 = b. 7 3 = 21 is equivalent to 7 = = c = 12 is equivalent to 23 = and 11 = and d = 3 is equivalent to 12 = 3 4 and 4 = a. x + 7 = 15 is equivalent to x = b. 7y = 21 is equivalent to y = c. w 11 = 12 is equivalent to w = d. n 4 = 3 is equivalent to n = a. A charm bracelet costs $24 plus $3 per charm. b. The point (17, 75) lies on the graph of the equation because 75 = (17); (60, 12) does not satisfy the equation. c. If you want to spend exactly $75 on a bracelet for your sister, how many charms can you afford? 60. a. The rental cost for a theater is $120 plus $4.50 per student. b. The point (8, 156) lies on the graph, because 156 = (8). (15, 180) is not on the graph, because (15). c. Suppose a class wants to spend no more than $156 on renting a theater. How many people could go? x x < 5.5 (See Figure 11.) > x (See Figure 12.) 63. x < 9 (See Figure 13.) 64. x > 6 (See Figure 14.) Figure 11 Figure 12 Figure 13 Figure 14 Variables and Patterns 6

7 Extensions 65. a. I = 20n b. F = 12n c. D = 200 d. S = 2.50n e. Both rules are correct because they both have the effect of subtracting expenses from income. f. By combining the various expenses into a simpler expression, one could get P = 20n ( n) or P = 20n n or some equivalent equation. Then, an even simpler form would be P = 5.50n 200. This form shows that each ticket brings income of $5.50 after food and cleanup costs are deducted. From the total income, $200 will be subtracted for the fixed DJ expense. You can know you are correct by substituting 100 into each equation to get $350 profit for each. 66. a. The relationship predicts that the business will lose one customer for every increase of $10 in the price. That pattern is shown explicitly in the table and by the downward slope in the graph of the relationship. (See Figures 15 and 16.) Figure 15 Price $50 $100 $150 $200 $250 $300 $350 $400 $450 $500 Customers Figure 16 Variables and Patterns 7

8 b. The relationship of tour income I to tour price p can be expressed with the equation I = ( p)p because total income equals number of customers times income per customer. c. The expression for calculating tour income in part (b) is equivalent to 50p 0.10p 2, because the Distributive Property has been applied and p(p) = p 2. d. Tables and graphs of the income relationship suggest that a maximum income of $6,250 occurs when the price is $250. (See Figures 17 and 18.) 67. a. d = 12t for times less than half an hour; d = (t 0.5) for times greater than half an hour b. d = 15(t 0.5) Figure 17 Price ($) Income ($) 2,250 4,000 5,250 6,000 6,250 6,000 5,250 4,000 2,250 0 Figure 18 Variables and Patterns 8

9 c. (See Figures 19 and 20.) d. Since the race will clearly last longer than half an hour, we ll solve using the equations for times greater than that amount: (t 0.5) = 15(t 0.5) has solution t = 1.7 hours, or one hour and 42 minutes. e. The lead older rider will catch up with the lead younger rider after 1.7 hours. This is shown by the crossing of graphs. 68. a. The card with 150 points actually offers only 12 (exactly 12.5) rides, while the card with 100 points offers 16 (exactly ) rides. b r = 100 6r when r = c. r < a. Sending 30 students costs $775. Sending 60 students costs $1,300. b. C = s c. 1,000 = s has solution s So the school can send 42 students and stay under the $1,000 limit. (Part students don t make sense in this situation, so the correct answer is 42.) The card with more starting points will appear to be the better deal until the ninth ride. Figure 19 Time in Race (hours) Distance Young Riders (miles) Distance Older Riders (miles) Figure 20 Variables and Patterns 9