Exponential and Logarithmic Modeling

Transcription

1 Exponential and Logarithmic Modeling Multiple Choice Identify the choice that best completes the statement or answers the question. 1. An initial population of 505 quail increases at an annual rate of 23%. Write an exponential function to model the quail population. a. b. c. d. 2. Write an exponential function for a graph that includes (1, 15) and (0, 6). a. b. c. d. 3. Find the annual percent increase or decrease that models. a. 230% increase b. 130% increase c. 30% decrease d. 65% decrease 4. The half-life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 10 days. Round your answer to the nearest thousandth. a. ; kg b. ; 0 kg c. ; kg d. ; kg 5. Suppose you invest $1600 at an annual interest rate of 4.6% compounded continuously. How much will you have in the account after 4 years? a. $ b. $6, c. $10, d. $1, How much money invested at 5% compounded continuously for 3 years will yield $820? a. $ b. $ c. $ d. $ The table shows some notable earthquakes that occurred in recent years. How many times more energy was released by the earthquake in Peru than by the earthquake in Mexico? Earthquake Location Date Richter Scale Measure Italy October 31, El Salvador February 13, Afghanistan May 30, Mexico January 22, Arequipa, Peru June 23, [Source: World Almanac 2004, p. 190] a. about 0.50 times as much energy b. about 15 times as much energy c. about times as much energy d. about 5.48 times as much energy

2 The ph of a liquid is a measure of how acidic or basic it is. The concentration of hydrogen ions in a liquid is labeled. Use the formula to answer questions about ph. 8. Find the ph level, to the nearest tenth, of a liquid with [H + ] about. a. 3.8 b. 3.8 c. 2.2 d A construction explosion has an intensity I of W/m 2. Find the loudness of the sound in decibels if and W/m 2. Round to the nearest tenth. a decibels b decibels c decibels d decibels 10. A company with loud machinery needs to cut its sound intensity to 37% of its original level. By how many decibels would the loudness be reduced? Use the formula. Round to the nearest hundredth. a decibels b decibels c decibels d decibels 11. Solve. a b c d Solve. a b c d Write the expression as a single natural logarithm. 13. a. b. c. d. 14. a. b. c. d. 15. The sales of lawn mowers t years after a particular model is introduced is given by the function y =, where y is the number of mowers sold. How many mowers will be sold 2 years after a model is introduced? Round the answer to the nearest whole number. a. 37,897 mowers b. 7,383 mowers c. 15,901 mowers d. 17,000 mowers 16. The generation time G for a particular bacteria is the time it takes for the population to double. The bacteria increase in population is shown by the formula, where t is the time period of the population increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the end of the time period. If the generation time for the bacteria is 6 hours, how long will it take 8 of these bacteria to multiply into a colony of 7681 bacteria? Round to the nearest hour.

3 a. 177 hours b. 76 hours c. 4 hours d. 85 hours 17. The amount of money in an account with continuously compounded interest is given by the formula, where P is the principal, r is the annual interest rate, and t is the time in years. Calculate to the nearest hundredth of a year how long it takes for an amount of money to double if interest is compounded continuously at 6.2%. Round to the nearest tenth. a. 1.1 yr b. 6.9 yr c yr d. 0.6 yr Short Answer 18. Without graphing, determine whether the function represents exponential growth or exponential decay. 19. Without graphing, determine whether the function represents exponential growth or exponential decay. 20. The exponential decay graph shows the expected depreciation for a new boat, selling for $3500, over 10 years Value($) years a. Write an exponential function for the graph. b. Use the function in part a to find the value of the boat after 9.5 years. 21. One method of dating artifacts is radiocarbon dating. An artifact s age y in years can be found by using the function, where x is the number of beta radiation emissions per minute per gram of carbon in the artifact. a. Use a graphing calculator to graph the function. Sketch the graph on the grid.

4 20000 y x b. Use your calculator to find the approximate value of y when x is 6. Essay 22. The table shows the number of squirrels in a particular forest t years after a forest fire. Number of Squirrels Years Squirrels a. Explain how the population of squirrels is changing each year. b. Write a function to model the situation. Explain what each number represents. 23. Suppose you invest $580 at 10% compounded continuously. a. Write an exponential function to model the amount in your investment account. b. Explain what each value in the function model represents. c. In how many years will the total reach $3600? Show your work. 24. The formula gives the average atmospheric pressure P in pounds per square inch, at an altitude x in miles above sea level. a. Find the elevation at which the average atmospheric pressure is 8.4 lb/in. 2. Show the steps you used to solve this problem. b. What is the average atmospheric pressure at sea level? Explain. Other

5 25. In a particular region of a national park, there are currently 330 deer, and the population is increasing at an annual rate of 11%. a. Write an exponential function to model the deer population. b. Explain what each value in the model represents. c. Predict the number of deer that will be in the region after five years. Show your work.

6 Exponential and Logarithmic Modeling Answer Section MULTIPLE CHOICE 1. ANS: D PTS: 1 DIF: L1 REF: 8-1 Exploring Exponential Models OBJ: Exponential Growth STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 2 KEY: exponential function growth factor 2. ANS: A PTS: 1 DIF: L1 REF: 8-1 Exploring Exponential Models OBJ: Exponential Growth STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 3 KEY: exponential function growth factor 3. ANS: B PTS: 1 DIF: L3 REF: 8-1 Exploring Exponential Models OBJ: Exponential Decay STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 6 KEY: exponential decay exponential function exponential growth percent 4. ANS: C PTS: 1 DIF: L1 REF: 8-2 Properties of Exponential Functions OBJ: Comparing Graphs NAT: NAEP A1e NAEP A1h CAT5.LV21/22.45 CAT5.LV21/22.50 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.DI IT.LV17/18.DP IT.LV17/18.PS S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.15 TV.LV21/22.17 TV.LV21/22.47 TV.LV21/22.52 TV.LVALG.56 STA: NC 2.03 TOP: 8-2 Example 3 KEY: exponential decay exponential function MSC: NAEP A1e NAEP A1h CAT5.LV21/22.45 CAT5.LV21/22.50 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.DI IT.LV17/18.DP IT.LV17/18.PS S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.15 TV.LV21/22.17 TV.LV21/22.47 TV.LV21/22.52 TV.LVALG ANS: D PTS: 1 DIF: L1 REF: 8-2 Properties of Exponential Functions OBJ: The Number e NAT: NAEP A1e NAEP A1h CAT5.LV21/22.45 CAT5.LV21/22.50 CAT5.LV21/22.53 IT.LV17/18.AM

7 IT.LV17/18.DI IT.LV17/18.DP IT.LV17/18.PS S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.15 TV.LV21/22.17 TV.LV21/22.47 TV.LV21/22.52 TV.LVALG.56 STA: NC 2.03 TOP: 8-2 Example 5 KEY: exponential function exponential growth interest rates problem solving the number e compounding continuously percent MSC: NAEP A1e NAEP A1h CAT5.LV21/22.45 CAT5.LV21/22.50 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.DI IT.LV17/18.DP IT.LV17/18.PS S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.15 TV.LV21/22.17 TV.LV21/22.47 TV.LV21/22.52 TV.LVALG ANS: D PTS: 1 DIF: L2 REF: 8-2 Properties of Exponential Functions OBJ: The Number e NAT: NAEP A1e NAEP A1h CAT5.LV21/22.45 CAT5.LV21/22.50 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.DI IT.LV17/18.DP IT.LV17/18.PS S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.15 TV.LV21/22.17 TV.LV21/22.47 TV.LV21/22.52 TV.LVALG.56 STA: NC 2.03 KEY: exponential function exponential growth interest rates percent problem solving the number e compounding continuously MSC: NAEP A1e NAEP A1h CAT5.LV21/22.45 CAT5.LV21/22.50 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.DI IT.LV17/18.DP IT.LV17/18.PS S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.15 TV.LV21/22.17 TV.LV21/22.47 TV.LV21/22.52 TV.LVALG ANS: D PTS: 1 DIF: L1 REF: 8-3 Logarithmic Functions as Inverses OBJ: Writing and Evaluating Logarithmic Expressions NAT: NAEP A2a NAEP A3a NAEP A3b CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.56 STA: NC 1.01 NC 2.01 TOP: 8-3 Example 1 KEY: problem solving evaluating logarithms logarithm MSC: NAEP A2a NAEP A3a NAEP A3b CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG ANS: C PTS: 1 DIF: L2 REF: 8-3 Logarithmic Functions as Inverses OBJ: Writing and Evaluating Logarithmic Expressions NAT: NAEP A2a NAEP A3a NAEP A3b CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.56 STA: NC 1.01 NC 2.01 TOP: 8-3 Example 4 KEY: logarithm problem solving MSC: NAEP A2a NAEP A3a NAEP A3b CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG ANS: C PTS: 1 DIF: L1 REF: 8-4 Properties of Logarithms OBJ: Using the Properties of Logarithms NAT: NAEP A2e CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.53 STA: NC 1.01 TOP: 8-4 Example 4 KEY: properties of logarithms problem solving MSC: NAEP A2e CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.53

8 10. ANS: D PTS: 1 DIF: L1 REF: 8-4 Properties of Logarithms OBJ: Using the Properties of Logarithms NAT: NAEP A2e CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.53 STA: NC 1.01 TOP: 8-4 Example 4 KEY: properties of logarithms problem solving MSC: NAEP A2e CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG ANS: A PTS: 1 DIF: L2 REF: 8-5 Exponential and Logarithmic Equations OBJ: Solving Logarithmic Equations NAT: NAEP A2b CAT5.LV21/22.50 CAT5.LV21/22.51 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.56 STA: NC 1.01 NC 2.01 TOP: 8-5 Example 7 KEY: logarithmic equation properties of logarithms MSC: NAEP A2b CAT5.LV21/22.50 CAT5.LV21/22.51 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG ANS: B PTS: 1 DIF: L2 REF: 8-5 Exponential and Logarithmic Equations OBJ: Solving Logarithmic Equations NAT: NAEP A2b CAT5.LV21/22.50 CAT5.LV21/22.51 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.56 STA: NC 1.01 NC 2.01 TOP: 8-5 Example 7 KEY: logarithmic equation properties of logarithms MSC: NAEP A2b CAT5.LV21/22.50 CAT5.LV21/22.51 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG ANS: D PTS: 1 DIF: L1 REF: 8-6 Natural Logarithms OBJ: Natural Logarithms NAT: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 STA: NC 1.01 NC 2.01 TOP: 8-6 Example 1 KEY: simplifying a natural logarithm properties of logarithms MSC: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/ ANS: D PTS: 1 DIF: L2 REF: 8-6 Natural Logarithms OBJ: Natural Logarithms NAT: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 STA: NC 1.01 NC 2.01 TOP: 8-6 Example 1 KEY: simplifying a natural logarithm properties of logarithms MSC: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/ ANS: D PTS: 1 DIF: L1 REF: 8-6 Natural Logarithms OBJ: Natural Logarithms NAT: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM

9 TV.LV21/22.12 STA: NC 1.01 NC 2.01 TOP: 8-6 Example 2 KEY: simplifying a natural logarithm logarithmic function problem solving MSC: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/ ANS: D PTS: 1 DIF: L2 REF: 8-5 Exponential and Logarithmic Equations OBJ: Solving Logarithmic Equations NAT: NAEP A2b CAT5.LV21/22.50 CAT5.LV21/22.51 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.56 STA: NC 1.01 NC 2.01 TOP: 8-6 Example 2 KEY: evaluating logarithms logarithmic equation properties of logarithms problem solving MSC: NAEP A2b CAT5.LV21/22.50 CAT5.LV21/22.51 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG ANS: C PTS: 1 DIF: L2 REF: 8-6 Natural Logarithms OBJ: Natural Logarithmic and Exponential Equations NAT: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 STA: NC 1.01 NC 2.01 TOP: 8-6 Example 5 KEY: exponential equation properties of logarithms problem solving MSC: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 SHORT ANSWER 18. ANS: exponential growth PTS: 1 DIF: L1 REF: 8-1 Exploring Exponential Models OBJ: Exponential Decay STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 4 KEY: exponential function exponential growth reasoning 19. ANS: exponential decay PTS: 1 DIF: L1 REF: 8-1 Exploring Exponential Models OBJ: Exponential Decay STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 4

10 KEY: exponential decay exponential function reasoning 20. ANS: a. Estimates for a and b may vary. Sample: b. about $250 PTS: 1 DIF: L1 REF: 8-1 Exploring Exponential Models OBJ: Exponential Decay STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 6 KEY: exponential decay exponential function graphing problem solving multi-part question 21. ANS: y a. b. about 6,827 years x PTS: 1 DIF: L3 REF: 8-3 Logarithmic Functions as Inverses OBJ: Graphing Logarithmic Functions NAT: NAEP A2a NAEP A3a NAEP A3b CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.56 STA: NC 1.01 NC 2.01 TOP: 8-3 Example 6 KEY: graphing logarithmic function multi-part question problem solving MSC: NAEP A2a NAEP A3a NAEP A3b CAT5.LV21/22.50 CAT5.LV21/22.55 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.13 TV.LV21/22.52 TV.LVALG.56

11 ESSAY 22. ANS: [4] a. Answers may vary. Sample: The population doubles each year. It appears that the population is increasing exponentially. b. A function for the situation is. The number 30 represents the initial population at 0 years. The number 2 represents the growth factor of 2. [3] one minor mathematical error or a reasoning error [2] poor explanation for part a. or error in the function model [1] only one part of the question answered PTS: 1 DIF: L2 REF: 8-1 Exploring Exponential Models OBJ: Exponential Growth STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 2 KEY: exponential function growth factor problem solving writing in math extended response rubric-based question 23. ANS: [4] a. b. In the model, the coefficient of e is 580, the original investment. The formula for continuously compounded interest uses the number e raised to the power rt, where r is the rate as a decimal, in this case 0.1, and t is the time in years. c. To find the number of years to reach $3600, substitute 3600 into the model. Dividing and rounding to the nearest year, t 18. The investment will reach $3600 in about 18 years. [3] one error in computation or incomplete explanation [2] two errors in computation or no explanation [1] one correct answer with no explanation PTS: 1 DIF: L3 REF: 8-6 Natural Logarithms OBJ: Natural Logarithmic and Exponential Equations NAT: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 STA: NC 1.01 NC 2.01 KEY: compounding continuously exponential function exponential growth extended response graphing interest rates percent problem solving the number e writing in math rubric-based question MSC: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM

12 TV.LV21/ ANS: [4] a. Substitute 8.4 for P. Divide each side by Take the natural logarithm of each side. Simplify. Divide each side by The elevation is about 2.7 miles above sea level. Use a calculator. b. The average atmospheric pressure at sea level is 14.7 lb/in. 2 because x is 0 and. [3] one mathematical error or one incorrect answer [2] two mathematical errors or one error and an incomplete explanation [1] one correct answer with no explanation PTS: 1 DIF: L3 REF: 8-6 Natural Logarithms OBJ: Natural Logarithmic and Exponential Equations NAT: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 STA: NC 1.01 NC 2.01 TOP: 8-6 Example 5 KEY: exponential equation properties of logarithms problem solving extended response rubric-based question writing in math MSC: NAEP A3a CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.52 IT.LV17/18.AM TV.LV21/22.12 OTHER 25. ANS: a. b. In the model, 330 represents the initial population of deer. The growth factor is represented by or c. To predict the number of deer present after 5 years, substitute 5 into the function and evaluate. function model Substitute 5 for x. Use a calculator. There will be about 556 deer in the region. PTS: 1 DIF: L2 REF: 8-1 Exploring Exponential Models

13 OBJ: Exponential Decay STA: NC 2.03 NC 2.03b NC 2.04 NC 2.04b TOP: 8-1 Example 2 KEY: exponential decay exponential function multi-part question percent problem solving writing in math