1ACE Exercise 22. Name Date Class

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1 1ACE Exercise 22 Investigation Many single-celled organisms reproduce by dividing into two identical cells. Suppose an amoeba splits into two amoebas every half hour. a. An experiment starts with one amoeba. Make a table showing the number of amoebas at the end of each hour over an 8-hour period. A table is started below. HINT Drawing a diagram may help to show what happens when cells divide. Use it to help you think about what goes on in the table. Start = 0 hours 1 2 hour 1 hour Hours Number of Amoebas Amoeba Reproduction

2 1ACE Exercise 22 (continued) Investigation 1 b. Write an equation for the number of amoebas a after t hours. Note that when t = 1, amoebas (a) = 4 when t = 2, amoebas (a) = 1 when t = 3, amoebas (a) = 4 a = c. After how many hours will the number of amoebas reach 1 million (a = 1,000,000)? HINT Extend your table or use your equation. d. Make a graph of the (time, amoeba) data from part (a). HINT Use the table to help you think about scales for the axes. 153

3 1ACE Exercise 22 (continued) Investigation 1 e. What similarities do you notice in the pattern of change for the number of amoebas and the patterns of change for other problems in this investigation? What differences do you notice in the pattern of change for the number of amoebas and the patterns of change for other problems in this investigation? 154

4 2ACE Exercise 3 (continued) Investigation 2 3. Leaping Leonora just signed a contract with a women s basketball team. The contract guarantees her $20,000 the first year, $40,000 the second year, $80,000 the third year, $10,000 the fourth year, and so on, for ten years. a. Make a table showing Leonora s salary each year of this contract. Leonora s Salary Year Salary $20,000 $40,000 $80,000 $10,000 Year Salary b. What total amount of money will Leonora earn over the 10 years? c. Describe the growth pattern in Leonora s salary. d. Write an equation for Leonora s salary s for any year n of her contract. s = 155

5 3ACE Exercise 1 Investigation 3 1. In parts of the United States, wolves are being reintroduced to wilderness areas where they had become extinct. Suppose 20 wolves are released in northern Michigan, and the yearly growth factor for this population is expected to be 1.2. HINT What is meant by growth factor? a. Make a table showing the projected number of wolves at the end of each of the first years. Year Number of Wolves (20 x 1.2) b. Write an equation that models the growth pattern of the wolf population W for any year n. W = c. How long will it take for the new wolf population to exceed 100 (W > 100)? 15

6 4ACE Exercise 1 Investigation 4 1. Latisha has a 24-inch string of licorice (LIK uh rish) to share with her friends. As each friend asks her for a piece, Latisha gives him or her half of what she has left. She does not eat any of the licorice herself. a. Make a table showing the length of licorice Latisha has left each time she gives a piece away. Latisha s Licorice Number of Friends Size of Licorice Left (in.) b. Make a graph of the data from part (a). 157

7 4ACE Exercise 1 (continued) Investigation 4 c. Suppose that, instead of half the licorice that is left each time, Latisha gives each friend 4 inches of licorice. Make a table and a graph for this situation. Latisha s Licorice Number of Friends Size of Licorice Left (in.) 20 1 d. Compare the tables and the graphs for the two situations. Explain the similarities. Explain the differences. 158

8 5ACE Exercise 31 Investigation Manuela said it must be true that 2 10 = because she can group as ( ) ( ). a. Verify that Manuela is correct by evaluating both sides of the equation =2 4 2 = HINT = 1,024 ( ) ( ) = 1 4 = 1, = 1,024 b. Use Manuela s idea of grouping factors to write three other expressions that are equivalent to Evaluate each expression you find to verify that it is equivalent to The first one has been done for you = ,024 = ,024 = 1, = = 159

9 5ACE Exercise 31 (continued) Investigation 5 c. The standard form for 2 7 is 128 (meaning 2 7 = 128), and the standard form for 2 5 is 32 (meaning 2 5 = 32). Use these facts to evaluate HINT What do you notice about the relationship between 2 7,2 5, and 2 12? HINT What is the standard form of 2 12? Explain your work. 10

10 5ACE Exercise 31 (continued) Investigation 5 d. Test Manuela s idea to see if it works for exponential expressions with other bases, such as 3 8 or (1.5) 11. Test several cases. HINT Test =, = ,51 = 81 81,51 =,51 Test 3 more cases with different bases Give an argument supporting your conclusion. 11

11 Unit Test 1. Several species of whale have been declared endangered. When the populations of a particular whale species fall dangerously low, biologists encourage governments to agree to a ban on hunting the species. Suppose that, in the year 2000, there were only 5,000 whales of a particular species and that the population was predicted to continue to decline as shown in the table. a. Which equation below models this population pattern? A. W = 5,000(0.1 y ) B. W = 5,000(0.9 y ) C. W = 5, y D. W = 5,000 y b. What is the decay factor for the relationship? Explain how you determined your answer. Year (y) 0 (2000) 1 (2001) 2 (2002) 3 (2003) 4 (2004) 5 (2005) (200) Whales (w) 5,000 4,500 4,050 3,45 3,281 2,952 2,57 c. According to the prediction, what will the whale population be in 2007? d. Suppose the danger point for these whales comes when the population falls below 2,000 whales. When will this happen? Explain. 12

12 Unit Test (continued) 2. a. On Grid I, sketch and label graphs of y = 2 x and y = 2.5 x. On Grid II, sketch and label graphs of y = 0.5 x and y = 0.8 x. 40 y Grid I 30 y Grid II x x 3 4 b. In Grid I, which equation represents the faster rate of growth? c. In Grid II, which equation represents the faster rate of decay? d. How does the graph help you to answer parts (b) and (c)? e. How do the equations help you to answer parts (b) and (c)? 13

13 Unit Test (continued) 3. Belinda has a plan for distributing prize money for a trivia contest. For the first correct response, the contestant will receive $500. For the second correct response, the contestant will receive an additional $100, for a total of $00. For the third correct response, the contestant will receive $100 more, for a total of $700. Belinda s plan continues in this pattern. a. Make a table showing the amount of money a contestant would receive for answering questions 1 through correctly. Trivia Contest Prize Money Number of Correct Responses Total Money Received $500 $00 $700 b. Make a graph of the data in your table. c. Write an equation for the relationship between the number of correct responses c and the amount of money the contestant will receive m. 14

14 Unit Test (continued) 4. Monty has a different plan for distributing prize money for the trivia contest. The contestant will receive $5 for the first correct response. For the second correct response, the total winnings will increase to $25, for the third correct response, the total winnings will increase to $125, and so on. a. Make a table showing a contestant s earnings for answering questions 1 through correctly. Trivia Contest Prize Money Number of Correct Responses Total Earnings $5 $25 $125 b. Make a graph of the data in your table. c. Write an equation for the relationship between the number of correct responses c and the amount of money the contestant will receive m. 15

15 Unit Test (continued) 5. How are the patterns of change in Belinda s and Monty s plans (Exercises 3 and 4) alike? How are they different?. Decide whether each of the following statements is true or false. Explain your reasoning. a = b c. (3 ) 8 = d. = 10 3 e. 7 0 = 1 7. Write in exponential form. 1

What is the growth factor? Explain how you found it. Growth of Elk Population. Population

What is the growth factor? Explain how you found it. Growth of Elk Population. Population Applications 1. In parts of the United States, wolves are being reintroduced to wilderness areas where they had become extinct. Suppose 2 wolves are released in northern Michigan, and the yearly growth