Exponential Modeling/Regression Name: 1) John decided to start investing for his retirement with the money he received when his grandfather passed away. John s grandfather passed away when he was 23 years old and gave John $20,000. John, being poor at the time but wanting to save money, simply placed the money in a savings account that compounded annually with a 5% interest rate. When John turned 36, he received a $50,000 financial windfall from a work-place accident. John took that money and the money in his savings account and placed both in a mutual fund that compounded semi-annually with an interest rate of 11.5%. John hoped to that he could live out his retirement from this mutual fund. If John wanted to retire and took out the money from the mutual fund at age of 67, how much money would he have? 2) Wounds heal at a certain rate which is given by the equation W = W o e -0.25t where W is the size of the wound after a certain amount of time, W o is the initial size of the wound and t is the amount of time in days. Initially the wound is 25 square millimeters. What would be the size of the wound after 4 days? 3) How much money should be invested at 5% compounded quarterly for 20 years so that you have $20000 at the end of the 20 years? 4) In the television show Futurama, the main character, Fry, awakened from being cryogenically frozen for 1000 years. The bank teller told Fry that he had 93 cents at the time he was frozen and that his account had an interest rate of 2.25%. Assuming the account compounded yearly, how much money does Fry have in his account now?
5) The population of Winnemucca, Nevada can be modeled by P = 6191(1.04) t where t is the number of years since 1990. What was the population in 1990? By what percent did the population increase each year? What is the population in 2016? 6) Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling. If we start with only one bacteria which can double every hour, how many bacteria will we have by the end of one day? 7) An adult takes 400 mg of ibuprofen every 4 hours to ease a headache. Each hour, the amount of ibuprofen in the person s system decreases by 39%. How much ibuprofen is in their system after 6 hours? 8) You buy stocks for Walmart, Google, and Mattel. You invest $1000 into Walmart, $1500 into Google, and $2500 into Mattel. Google and Walmart both compound quarterly whereas Mattel is semiannually. Over the course of two years Walmart s value increased by 7.5%, Google increased by 12%, and Mattel decreased by 13.3%. How much money did you make (or lose) at the conclusion of those two years (above or below the original $5000 you invested initially)? 9) Caffeine has a half-life of about 5 hours. This means the amount of caffeine in your body will be halved after 5 hours. You drink a venti size pumpkin spice coffee from Starbucks (150 mg caffeine) at 8am, an 8oz Dr. Pepper can (41 mg caffeine) at noon, and eat a chocolate bar (9 mg caffeine) at 4pm. How much caffeine is in your system at 10pm? (Pro-tip: you will use fractions or decimals as exponents)
10) What is the lowest interest rate you could use to be able to turn $10,000 into $75,000 in less than 10 years with an account that compounds quarterly? (Pro-tip: don t forget to turn your percentage rate into a decimal for your equation) 11) A chemist and a biologist want to test if a certain chemical is effective in controlling a particular bacteria. A specific colony of this bacteria satisfies the exponential growth law P(t) = 100(4.5) t, where t is in hours. At time t = 0 the two scientists expose the colony to the chemical which they hope will control the bacteria. The biologist, at hourly intervals, calculates the number of bacteria. Her data is tabulated below. Do you think the chemical was effective in controlling the bacteria? 12) Let p(t) = 200e kt represent the number of bacteria in a petri dish after t days. Suppose the number of bacteria doubles every 5 days. What must k equal? 13) A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial. Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth. Use the equation to predict how many coins would be returned to the box after the eighth trial.
14) The population growth of Boomtown is shown in the accompanying graph. If the same pattern of population growth continues, what will the population of Boomtown be in the year 2020? 15) Jean invested $380 in stocks. Over the next 5 years, the value of her investment grew, as shown in the accompanying table. Write the exponential regression equation for this set of data, rounding all values to two decimal places. Using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase. 16) The accompanying table shows the number of bacteria present in a certain culture over a 5-hour period, where x is the time, in hours, and y is the number of bacteria. Write an exponential regression equation for this set of data, rounding all values to four decimal places. Using this equation, determine the number of whole bacteria present when x equals 6.5 hours. 17) The half-life of a radioactive material is 50 years. How long will it take a 10 kg sample to decay to 5 kg? 2.5 kg? 18) After 5 years, 20 grams of a radioactive material has decayed to 5 grams. What is the half-life?
19) A chemist has a 250-gram sample of a radioactive material. She records the amount remaining in the same table every day for a week and obtains the following data. Use a linear, quadratic, and exponential regression to estimate the amount of material left after 10 days. Which regression models give a reasonable answer? Justify your answer. Day Weight (grams) 1 250 2 208 3 158 4 130 5 102 6 80 7 65 20) Compare the rate of change in the pair of functions in the graph by identifying the interval where it appears f(x) is changing faster and the interval where it appears g(x) is changing faster. Write a function that could represent the exponential equation and use it to find the points of intersection between g(x) and f(x).