Exponential Models and Sequences: MMAT 3320 Notes - Prof. Roger A. Knobel, Study notes of Mathematics

Download Exponential Models and Sequences: MMAT 3320 Notes - Prof. Roger A. Knobel and more Study notes Mathematics in PDF only on Docsity! MMAT 3320 NOTES SECTION 5 PAGE 1 5 EXPONENTIAL MODELS A geometric sequence is a sequence with a recurrence relation given by an = ran−1, where r is a constant (called the common ratio). Assuming the starting index of this geometric sequence is 0, we have a1 = r⋅a0 a2 = r⋅a1 = r⋅r⋅a0 a3 = r⋅a2 = r⋅r⋅r⋅a0 … an = r⋅r r⋅…⋅r = a0⋅r n A model based on a geometric sequence is called a discrete exponential model since the isolated points of the graph of an = a0⋅r n lie on an exponential graph. A continuous exponential model is one with a continuous independent variable and has equation y = y0r x. Sometimes the last equation is written in the form y = y0e kx. EXAMPLE 1. Your savings account currently has $2300 in it. If money grows at 5% compounded annually, how much will be in your account after 9 years? a. Set up a recurrence relation with initial condition which models this problem. Let an denote the amount in dollars in your saving account n years from now. b. Solve the recurrence relation with initial condition in part a. MMAT 3320 NOTES SECTION 5 PAGE 2 c. Answer the question using the result of part b. d. The spreadsheet below shows the amount in the account after each year. Explain the discrepancy between the answer in part c and the answer in the spreadsheet. Number of years Amount in account 0 $2,300.00 1 $2,415.00 2 $2,535.75 3 $2,662.54 4 $2,795.67 5 $2,935.45 6 $3,082.22 7 $3,236.33 8 $3,398.15 9 $3,568.06 EXAMPLE 2. According to an internet site, the 2000 Census recorded 569,463 residents in the Hidalgo County/metro area, a 48.5 percent rise over the 383,545 residents in 1990. a. Let pn denote the population in the Hidalgo County/metro area n years after 1990. Letting r denote the annual growth rate, we have the recurrence relation pn = r⋅pn−1. Find the two initial conditions. MMAT 3320 NOTES SECTION 5 PAGE 5 b. A fossil is found in which 89% of the original carbon 14 has decayed. Estimate how long ago that the creature died. Round to the nearest hundred years. MMAT 3320 NOTES SECTION 5 PAGE 6 EXAMPLE 4. The take for a movie at the box office during each of the first six weeks of release is given in the table below: Number of weeks since release Revenue in millions of dollars 1 90.2 2 51.3 3 29.6 4 19.6 5 14.3 6 8.2 a. Which is better−a linear model or an exponential model? b. Use the better model to answer the following: If the movie studio plans to withdraw the film from theaters when the box office revenues drop to $250,000 per week, how many weeks will the movie run in theatrical release? MMAT 3320 NOTES SECTION 5 PAGE 7 HOMEWORK 1. If you deposit $750 at 4% compounded annually, a. how much money is in your account after 3 years? b. how long does it take until your account has $1200? c. how much interest does your money earn during its first 5 years? d. how much interest does your money earn during its first 10 years? 2. You borrow $5000 and must pay back $7800 in 6 years. What annual compound rate of interest are you being charged? Express as a percentage rounded to 2 decimal places. 3. The number of bacteria in a colony triples every day. Find the hourly growth rate. Express as a percentage rounded to 2 decimal places. 4. The population of a town was 13900 in 1970, 15000 in 1980, 16200 in 1990, and 18900 in 2000. a. Use exponential regression to find an exponential equation expressing the population P in terms of t, the number of years since 1970. Round values in equation to 5 decimal places. b. To the nearest hundred, predict what the population of the town will be in the year 2010. c. To the nearest year, find the doubling time of the population; in other words, find how long it takes for the population to double. 5. A certain radioactive substance has decayed to 90% of its original amount after 47 years. a. What percentage of the original substance remains after 67 years? Round to the nearest percentage point. b. Find the half-life of this substance. Round to the nearest year. 6. A study found that athletes who rested after exhausting exercise had a half-life of lactic acid removal of about 23 minutes. a. How much lactic acid has been removed after 10 minutes of rest? Round to the nearest percentage point. b. How long would it take to remove 85% of the lactic acid? Round to the nearest minute.