VIEWS: 17 PAGES: 4 POSTED ON: 9/16/2012 Public Domain
MAP4C Exponential Growth & Decay Modeling exponential growth and decay. Three familiar models (functions) that can represent growth are: Linear (y=mx+b) Quadratic (y=ax2+bx+c) Exponential (y=a(bx)) - simple interest - speed of a skydiver after - growth of bacteria - salary based on an jumping out of a plane - compound interest hourly wage Often, we only look at the first quadrant when using these function to model growth, as the x axis usually represents something like time which can’t be negative. Exponential growth: When looking at a population that grows exponentially, you either know information about the doubling period or the growth rate. Examples: a) A bacterial strain doubles in size every 3 minutes. If there are 1000 bacteria present initially, how many will there be in 15 minutes? This question involves doubling period. The doubling period is three. b) A valuable stamp was purchased for $6000 in the year 2000. The stamp appreciates in values at 6% per year. How much will it be worth in 40 years? This question involves growth rate. The value of the stamp is growing at a rate of 6%. y Ab x where; y represents the total amount (or population) after time t. A represents the amount after t=0. This is the initial amount. b represents the doubling period or growth rate. x represents the time. Page 1 of 4 MAP4C Exponential Growth & Decay Solutions to above examples. y Ab where A=1000, x is a time divided by 3. x a) In this problem we use the formula, 15 At =1000 2 3 =32 000 after 15 minutes there will be 32 000 bacteria. b) In this problem we use the formula y Ab , where A=6000, b=1.06* and x=40. x At 6000106 40 . after 40 years the stamp is worth $61 714.31. 61714.31 * Note that even though the growth rate is 6%, we use b=1.06 instead of just 0.06. You will recall that you also do this in the compound interest formula. This is so that the new amount is 6% more than the old amount, not just 6% of the old amount. Exponential Decay: Exponential decay is the model of a population that decreases in size. Graphically it looks likes this: This situation occurs with radioactive material which loses mass over time. Such problems will refer to the half life of a certain substance. Another situation that can be modeled by exponential decay is the depreciation of a car, or other purchase that loses value over time. y Ab x where; b represents the growth rate. Everything about this formula is exactly the same as for exponential growth with the following exception. If a population is growing exponentially at a rate of 6%, we say that b=1.06. It makes sense that the b value has to be bigger than one since the population is growing. If a value or population is decreasing at 6% then you could say it will only be 94% of the original amount, so the b value is b=0.94. Page 2 of 4 MAP4C Exponential Growth & Decay Examples: a) The half life of iodine-131 is 8 days. This means that whatever mass of iodine you have now, there will be half as much in 8 days. A scientist has 320 mg of I-131 which he stores for 40 days. If the half life is 8 days, how much will be left at the end of the 40 days? t 1 h At Ao 2 40 1 8 320 2 5 1 320 after 40 days, 10 mg of I-131 remain. 2 1 320 32 10 b) Mr. Ghobril buys a motorcycle for $16 000. Once he drives off the lot, the value of the motorcycle depreciates at a rate of 12% per year. How much will the bike be worth in 6 years? Note: Since the depreciation rate is 12%, that means that the bike is only worth 88% of its previous value from one year to the next. Therefore b=0.88. At c b t 16000 0.88 6 in 6 years Mr. Ghobril’s motorcycle will be worth $7430.47. 16000(0.464404086) 7430.47 Page 3 of 4 MAP4C Exponential Growth & Decay Homework In solving the following questions you should include a formula, show all your work and include a concluding statement with appropriate units. 1. A certain bacterial strain divides every hour producing two bacteria from every existing one (ie doubling!). If there are 200 bacterial in the culture, how many will there be in 8 hours? [ 51200 ] 2. A culture of cells is allowed to divide and after 4 hours there are now 10 000 cells. If the number of cells doubles every 2 hours, how many cells were there in the culture originally? How many will there be in 13 hours? [ 2500 , 226 274 ] 3. The population of a certain city is 810 000 and it is increasing at a rate of 4% per year. What will be the population in four years? [ 947 585 ] 4. A river is stocked with 5000 salmon. The population increases at a rate of 7% per year. What will the population be in 15 years? How many years does it take for the population of salmon to double? [ 13 795 , between 10 & 11 years , 10.24 ] 5. An element is decaying at a rate of 12%/h. Initially there were 100g. When will there be 40 g left? [ between 7 & 8 , 7 hrs. & 12 min. ] 6. The half life of crazium is 1620 years. If a mad scientist has 10 mg initially, how much would she have left in 50 years? Round your answer to the nearest thousandths. [ 9.788 mg ] 7. A science student produced 160 mg of radioactive sodium. She found that 45 h later there was only 20 mg left. What is the half life of radioactive sodium? [ 15 hours ] Page 4 of 4