I"! w w Introduction In this unit, you explored the properties and characteristics of exponential E and logarithmic (5 functions. You also saw numerous realworld applications ofthese functions, Including exponential growth and decay E , compounding Interest, and the Richter scaler which measures the magnitude of earthquakes. In this activity, you'll extend your exploration of exponential and logarithmic functions by researching the history behind the number I? and the natural logarithm. This knowledge will help you better understand how to use them to solve realworld problems. The two tasks In this activity Include: 1. An Investigation of the number e and the natural logarithm. 2. Applications of exponential and logarithmic functions. You may need to print some pages to work on these activities, but you can also work directly onllne for most ofthe steps. To sketch number lines, you can download this software 3 . Symbols, such as less than or greater than, needed to write equations can be found in MS Word 3. You can use the equation editor (5 to type In other parts of equationsr such as fractions. Evaluation Your teacher will use these rubrics to evaluate the completeness of your work as well as the clarity of thinking you exhibit. Total Points: 10 Task 1: Investigate the Number e and the Natural Logarithm Task Points: 3 Research and present the history of e. Address: . when e was discovered 0.5 . how the value of e is calculated . why e and the natural logarithm are used 0.5 . two specific real-world scenarios Task 2: Apply Exponential and Logarithmic Functions Task Points: 7 a. Given a town's population and population growth rate: 1. Determine the growth factor. 0.25 2. Write an equation to model growth. 0.5 3. Use the equation to estimate future population. 0.25 b. Given the price of a new car and its depreciation rate: 1. Determine the decay factor of the car's value. 0.25 2. Write an equation to model the value. 0.52. Write an equation to model growth. 0.5 3. Use the equation to estimate future population. 0.25 b. Given the price of a new car and its depreciation rate: 1. Determine the decay factor of the car's value. 0.25 2. Write an equation to model the value. 0.5 3. Use the equation to find the car's value after 6 years. 0.25 c. Given the initial deposit and annual interest rate, find the value after 10 years if interest is compounded: 1. annually 0.25 2. monthly 0.25 3. continuously 0.5 d. Use the half-life of radium-226 to determine the amount of a 50-gram sample that remains after 100 years. 1 e. Use the formula for magnitude of an earthquake to: 1. Find the magnitude of the 2010 Haiti earthquake. 0.5 2. Determine the difference in Richter scale readings for two earthquakes. 0.5 f. Use an equation modeling the spread of a flu virus to determine: 1. the number of students infected after 8 days 0.5 2. the number of days it takes to infect half the student population 0.5 g. Given the US population in 2000 and 2010: 1. Find the 10-year continuous growth rate. 0.5 2. Write an equation to model US population growth and estimate the population in 2020. 0.5Print Apply Exponential and Logarithmic Functions In this task, you will use your knowledge of exponential and logarithmic functions to solve real-world problems. Question 1 In 2006, the population of Tewksbury, Rhode Island was 25,000, and it was growing at an annual rate of 2.2%. Part A What is the growth factor for the town? BIU X X 2 15px AvLv V BV Characters used: 0 / 15000Question 2 Kelly bought a new car for $20,000. The car depreciates at a rate of 10% per year. Part A What is the decay factor for the value of the car? BIU X X 2 15px V v V BV Characters used: 0 / 15000 Part B Write an equation to model the car's value. BI U X X 2 15px v Avbv VPart C Use your equation to determine the value of the car six years after Kelly purchased it. BIU X X 2 15px v v Characters used: 0 / 15000 Question 3 Rachel invests $1,000 in a bank account that pays 5% annual interest. Part A How much money will Rachel have in 10 years if the interest is compounded annually? BIU X X 2 15px v vPart B How much money will Rachel have in 10 years if the interest is compounded monthly? BI U X X 2 15px V AvLV Characters used: 0 / 15000 Part C How much money will Rachel have in 10 years if the interest is compounded continuously? BIU X X 2 15px v AVE MEE JEEZ V' E v Characters used: 0 / 15000Question 4 Radium-226, an isotope of radium, has a half-life of 1,601 years. Suppose your chemistry teacher has a 50-gram sample of radium-226. How much of the sample will be remaining after 100 years? BIU X X 2 15px V v E V B v Characters used: 0 / 15000 Question 5 On the Richter scale, the magnitude, M, of an earthquake is given by M = log ,, where E is the energy released by the earthquake measured in joules, and Eo is the energy released by a very small reference earthquake. Eo has been standardized to 10*$ joules. Part A The earthquake in Haiti in 2010 released 2.0 . 10 joules. What was its magnitude on the Richter scale? BIU X X 2 15px V AVL vPart B If the energy release of one earthquake is 10,000 times that of another, how much larger is the Richter scale reading of the larger earthquake than the smaller? BIU X X 2 15px v AVLV Characters used: 0 / 15000 Question 6 The spread of a flu virus on a college campus is modeled by y = 14 4gag on , where y is the number of students infected after t days. 5,000 Part A How many students are infected after 8 days? BIU x X 2 15px v Avv VPart B How many days will it take before half of the student population is infected? BIU X X 2 15px V AVE v E Characters used: 0 / 15000 Question 7 In the year 2000, the United States had a population of about 281.4 million people; by 2010, the population had risen to about 308.7 million. Part A Find the 10-year continuous growth rate using P = Pert. BIUX X 2 15px v VPart B Write an equation to model the population growth of the United States, and use it to estimate the population in 2020. BIU X X 2 15px A Characters used: 0 / 15000