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Exponential Functions, Worksheet 2 Classwork Objective: The main purpose of this activity is to apply your knowledge of exponential functions to model situations in the real world. By the time you complete the activity, you should be able to write a formula for an exponential function to fit the situation and solve equations in order to answer questions about the situation. Vocabulary: Exponential growth refers to something which increases by a constant ratio over regular intervals. Exponential decay refers to something which decreases by a constant ratio over regular intervals. 1) A car’s book value is an estimate of a car’s resale value. In 2005, Tom bought a used car for $7000, its book value at that time. Three years later, the car’s book value was $3584. Suppose the book value decreases exponentially. a) Find an equation to model the car’s book value over time. It will work best if you use fractions rather than decimals to compute the ratio. b) What is the book value of the car in 2011? c) Tom’s car was a 2002 model. What was the car’s value when new? 2) Here is something I pulled out of textbook on bacteriology: “The time interval required for a bacterial cell to divide or for a population of bacterial cells to double is called the generation time. Generation times for bacterial species growing in nature may be as short as 15 minutes or as long as several days.” So let’s say you are in medical school and you start out with a sample of bacteria that has 5000 little critters in it. Your particular species has a generation rate of 30 minutes. How long will it take for you to have over a million bacteria? 3) How many bacteria will there be after a week? How about 30 days? 4) Do you think that exponential growth is a reasonable type of model to use for something like bacteria? Discuss this with your group. By the way, your homework one day next week is going to be to look for some real-world data on something that you think might be exponential. It should be something you are interested in. So you might start thinking now about what type of thing you would like to gather data on. For example, is the popularity of Twitter growing exponentially? 5) When you deposit money in a bank account, you receive interest on the money in the account. The bank is paying you for the use of your money, which they in turn lend to other people. Interest rates are expressed as percentages, so if an account is paying 5% interest, then you will earn $5 for every $100 in the account, for each year that they have your money. a) Suppose you deposit $600. At the end of the year, how much interest will you get? b) How much interest will you get at the end of two years? 6) Actually, question b is kind of hard to answer. First we have to figure out what happened to the $30 interest that you earned at the end of the first year. Did you spend it or did you leave it in the bank account? That makes a difference. If you leave it in the bank, then at the end of the second year, you are getting interest on $630, not just interest on $600. So instead of $30, how much do you get? 7) Make a chart to show what happens if, each year, you leave the interest earned in your account and allow it to grow. How much money is in the account after ten years? Remember that the original investment, also known as the “principal”, was $600. How much total interest did you earn over 10 years? 8) How much interest would you earn over ten years if you left the $600 in the account, but spent the interest as soon as you received it? 9) Write an equation for an exponential function v(t) that models the growth in the value of your bank account over time. Test the function and make sure that v(10) equals the same thing as you see in your chart. If v(10) comes out as a very small number, then you made a mistake in identifying the ratio for this function. 10) So if you leave all of the money in the bank instead of spending the interest, you will be rewarded with an extra $77 over ten years. This is known as “compound interest”—it essentially means getting interest on your interest. It means that you are really earning better than 5% on your money. Let’s see how much better. You got approximately $377 over ten years, so that is about $37.70 per year. If I take $37.70 as a percentage of $600, it’s 37.7/600 ≈ .063. So that is better than 5%. It’s a little over 6% in fact. That rate is known as the “APY” of a bank account, which stands for “Annual Percentage Yield.” Suppose you leave the money in your account for twenty years instead of ten years. Does the APY change, or is it the same as leaving it in for ten years? 11) Different types of bank accounts have options on how often the money compounds. The example given above compounds annually. Other accounts compound more often. Let’s see what difference that might make. One option is compounding quarterly—four times a year. Here’s how it works. At the end of three months, the bank would see the original $600 in your account and they would say, “Oh, this would earn $30 worth of interest at the end of the year, so let’s give this nice student ¼ of that interest right now.” So, you now have $607.50 in your account at the end of three months. How much will you have at the end of six months? Check with me before you continue. 12) Make a new table of values to show what happens if interest in your account is compounded quarterly over a period of two years. How much do you have at the end of two years? 13) Come up with a formula for a function that describes the growth in your account when interest is compounded quarterly. Use it to see if you have more money at the end of ten years than you did when it was compounded annually. 14) Now you will experiment to see what happens as you change the way the interest is compounded. You can divide the labor in your group. One person should look at compounding the interest monthly, another daily, and another hourly. Each person should generate a data table for 10 rows and then develop a formula that fits the data. Once you have the formula, check to see how much money is in your account after ten years with your particular compounding rate. You should also compute the APY for your situation. 15) Clearly, the more frequently you compound the interest, the better it is for the account holder. Now try compounding the interest every second. Is it significantly different than compounding it every hour? 16) If it were possible to compound the interest continuously, would that mean that you could theoretically get an infinite amount of interest? 17) Suppose you are looking for an investment that will double in value over 15 years. What interest rate would you need to find if it is compounded daily? 18) By the way, when I put 5% up there in those problems about interest rates, that was wishful thinking. I opened an account about 7 years ago that started out earning 5%, but now it’s only earning 1.15%. Interest rates are really low right now. Some accounts are actually less than 1%. How long will it take for my money to double at the rate of 1.15% compounded daily? 19) Here is some data I found on global population growth. Year World Population 1 150 million 1350 300 million 1700 600 million 1800 900 million 1900 1.6 billion 1950 2.4 billion 1985 5 billion 2020 8 billion Based on this data, do you think that population is in fact growing exponentially? 20) Below is some information about the country of Liberia, taken from the CIA Factbook Based on this information, what will be the approximate population of Liberia in 2020? What assumptions do you have to make in order to answer that question? Can you tell from the other data how they computed the birth rate? Population: 3,685,076 (July 2010 est.) country comparison to the world: 129 Age structure: 0-14 years: 44.1% (male 760,989/female 758,554) 15-64 years: 53% (male 904,770/female 920,704) 65 years and over: 2.8% (male 47,013/female 49,760) (2010 est.) Median age: total: 18.4 years male: 18.3 years female: 18.4 years (2010 est.) Population growth rate: 2.782% (2010 est.) country comparison to the world: 19 Birth rate: 38.14 births/1,000 population (2010 est.) country comparison to the world: 18 Death rate: 10.88 deaths/1,000 population (July 2010 est.) country comparison to the world: 41 Net migration rate: 0.56 migrant(s)/1,000 population (2010 est.) country comparison to the world: 61