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1 Applications of Exponential Growth and Decay Name Exponential functions: y x = ab is the parent function, growth if ; decay if. On the graph of the function, the a represents the y-intercept. This is often referred to as the initial amount. We now will encounter problems that give us the initial amount and a rate of increase or decrease. If you are given the percent of increase or decrease and an initial amount use one of the following forms to find the growth or decay function. The rate and time will be for the same period: year, month, day, etc. Growth: y = a(1 + r) t (1 + r) is the growth factor a is the initial amount r is the percent expressed as a decimal Decay: y = a(1 r) t (1 r) is the decay factor t is time 1. Write a model that represents the amount of widgets you will have in t months if the initial amount is 50 widgets and the rate of increase is 15% each month. 2. Write a model that represents the amount of widgets you will have in t years if the initial amount is 1000 widgets and the rate of decrease is 6% each year. 3. In 2000, the average price of a football ticket for a Chicago Bears game was $ During the next 4 years, the price increased an average of 6% each year. a) Write a model giving the average price p (in dollars) of a ticket t years after b) Graph the model. LABEL THE AXES. What is the y-intercept? c) Estimate the year when the average price of a ticket was about $ d) If the pattern continues, what will be the price of a ticket in 2016?

2 4. You buy a mountain bike for $200. The value of the bike decreases by 25% each year. Write the model for the value of the bike after t years, and then estimate the value of the bike after 5 years. 5. Vincent Van Gogh s painted Irises in 1889 which was auctioned for $53.9 million dollars in It had previously been purchased for $80,000 in Assuming exponential growth, by what percent did its value increase each year? 6. In 1992, 1219 monk parakeets were observed in the United States. For the next 11 years, about 14% more parakeets were observed each year. Write the model for the number of monk parakeets after t years, and then estimate the year when the number of monk parakeets total 25,000 if the pattern continues. 7. A tool and die business purchases a piece of equipment for $250,000. The value of the equipment depreciates at a rate of 12% each year. Identify the following: Initial amount annual % decrease decay factor a) Write a model for the value of the equipment after t years. b) Graph the model. What is the y-intercept? c) What is the value of the equipment after 5 years? d) Use your model to estimate when the equipment will have a value of $70,

3 DAY 2 Problems that involve compound growth, such as interest and banking, use the following exponential growth function. The rate will be for a year, however the interest will be compounded n times per year. (this is the same as the growth function from the first page if the rate is compounded yearly) A= P 1+ r n nt P = principal or beginning amount r = rate of interest n = frequency of compounding in one year t = number of years A = the amount in the account in t years Fill in the following table for n given the number of times the rate is compounded per year. # of times yearly semi-annually quarterly monthly weekly daily n 1 6. You deposit $1500 in an account that pays x% interest. After t years how much money is in the account? x% 4.5% 5% 5¼ % Model using n and t.045 A = n nt t=15 years, if compounded Yearly $ $ $ Quarterly $ $ Monthly $ $ Daily $ $ You have $2000 in your bank s savings account, which is compounded monthly at a 6% interest rate. How many years would it take to double your money to $4000? 2. You have $5000 in your bank s savings account, which is compounded monthly at a 6% interest rate. How many years would it take to double your money to $10,000? -3-

4 3. You receive $200 that you want to invest in a savings account. After 25 years, how much difference would there be in the amount of the money in your savings account if you invest at Bank of America, currently paying 0.20% annual interest compounded daily compared to BB&T, currently paying 0.50% annual interest compounded daily? 4. You receive $200 that you want to invest in a savings account. After 25 years, how much difference would there be in the amount of the money in your savings account if you invest at Bank of America, currently paying 8.20% annual interest compounded daily compared to BB&T, currently paying 8.50% annual interest compounded daily? Is this the same difference as in problem #3? 5. How much must you deposit in an account that pays 8% annual interest, compounded quarterly to have $1000 after 5 years? 6. Find the interest rate that you would need to find at a bank that compounds daily if you wanted to double your money in 15 years? -4-

5 CONTINUOUS Growth and Decay (uses natural logs and e) rt P initial amount A = Pe A amount in account in t years r rate of interest t time in years 7. You invest $1500 in an account that pays 1.5% yearly interest. a) If your bank compounds interest monthly, how much money would you have after 10 years? b) If your bank compounds interest continuously, how much money would you have after 10 years? Applications DAY 3 More continuous growth/decay What problems would involve continuous growth?? kt a is the initial amount y = ae k is the rate of growth or decay if k > 0, growth; if k < 0, decay t is time 1. The number of bacteria in a certain population increases continuously. You have a sample of 2600 bacteria from this population. Assuming the continuous growth rate per hour is 4.5%, find the amount of time it takes for the bacteria to grow to a size of The element radium-226, a common isotope of radium, has a half-life of 1620 years. Find the rate of decay for radium

6 3. Archeologists uncover an ancient wooden tool. They analyze the tool and find that it has 22% as much carbon- 14 compared to the likely amount that it contained when it was made. Given that the half-life of carbon-14 is about 5730 years, about how old is the artifact? R ound your answer to the nearest 100 years. 4. In an over-fished area, the catch of a certain fish is decreasing exponentially. Use k = to determine how long will it take for the catch to reach half of its current the amount. 5. How many days will it take a culture of bacteria to increase from 2000 to 50,000? Use k = Cobalt, an element used to make alloys, has several isotopes. One of these, cobalt 60, is radioactive and has a half-life of 5.7 years. Cobalt 60 is used to trace the path of nonradioactive substances in a system. What is the value of k for cobalt 60? -6-

Simple Interest Formula

Simple Interest Formula Accelerated Precalculus 5.7 (Financial Models) 5.8 (Exponential Growth and Decay) Notes Interest is money paid for the use of money. The total amount borrowed (whether by an individual from a bank in the