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Exponential Growth and Decay Notes
You are going to work for me for one month; there are 2 options from which you can choose to be paid.
Which one would you choose?
Option 1: $1,000 a day for 31 days or
Option 2: $.01 on day 1,
$.02 on day 2,
$.04 on day 3,
$.08 on day 4, etc.? Justify your response.
Bacteria reproduce, or grow in number, by dividing. The total number of bacteria at a given time
is referred to as the population of bacteria. When each bacterium in a population of bacteria
divides, the population doubles.
You will model the growth of 25 bacteria; assume the entire population doubles every hour.
1. Complete the table below.
2. Write an algebraic expression that represents the population of bacteria after x
hours. (Hint: Factor out 25 from each population figure.) 3. Use your algebraic expression to find the population of bacteria after 10 hours and after 20 hours. 4. Suppose that the initial population of bacteria was 75 instead of 25. Find the population after 10 hours and after 20 hours. 5. Now suppose that the bacteria whose initial population was 25 triples every hour. Find the amount of bacteria after 4 hours and after 6.5 hours. 6. Now suppose that the bacteria population from #5 doubles every 20 minutes. Find the amount of bacteria after 6 hours and after 10 hours.
1. Need to find the multiplier (base) for exponential growth
-- Add growth rate (%) to 100% (then convert to a decimal)
is the number of growth periods
3. Initial amount times the multiplier raised to the number of growth periods
1. The population of the United States was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade.[Source: U.S. Census Bureau] Predict the population for the years 2000, 2010, and 2025. Assume that the growth rate remains a constant 8% for all decades. 2. Suppose that you invested $1000 in a company’s stock at the end of 1999 and that the value of that stock increased at a rate of about 15% per year. Predict the value of the stock, to the nearest cent, at the end of the years 2004 and 2009. 3. If a population’s growth rate is 1%per year, what is the population’s growth rate per decade? Exponential Decay
Can be thought of as a negative growth rate Subtract rate of decay from 100% (convert to a decimal)
Find the multiplier: 1. 2% rate of decay
1. The rate at which caffeine is eliminated from the blood stream of an adult is about 15% per hour. An adult drinks a caffeinated soda, and the caffeine in his or her bloodstream reaches a peak of 30 milligrams. Predict the amount, to the nearest tenth of a milligram, of caffeine remaining 1 hour after the peak level and 4 hours after the peak level. 2. Suppose you buy a car for $35,000 and its value decreases at a rate of about 8% per year. Predict the value of the car, to the nearest cent, after 4 years and after 6 years (after it’s paid for, Yeah!). 3. A certain medication is eliminated from the bloodstream at a rate of about 12% per hour. The medication reaches a peak level of 40 milligrams. Predict the amount, to the nearest tenth of a milligram, of the medication remaining 2 hours after the peak level and 10 hours after the peak level.
Newspaper Investigation: Goal: to explore situations in which quantities repeatedly double or split in half. Step 1. Unfold a sheet of newspaper, calculate its area, and record the area in Table 1 in the first row.
Step 2. Fold the sheet of newspaper in half. Tear along the fold so that you have 2 pieces. What is the area of each piece? Record that information in Table 1. Fill in Table 2. Now stack the pieces of newspaper on top of each other. Fold the stack in half and tear along the fold. Fill in the tables. Predict how many times you will be able tear the stack of newspaper: ____________________. (You must tear all sheets at the same time) Continue stacking, folding, and tearing the newspaper until you cannot tear the stack again. Extend Tables 1 and 2 as needed to record the number of pieces and the area of each piece. 3. Suppose you could tear the newspaper stack as many times as you predicted above; how many pieces would you have? What would be the area of one piece? 4. Suppose a single piece of newspaper measures .0032 inches thick. How tall would the stack be after 8 tears? 10 tears? 20 tears? 5. Write an expression in terms of n
tears for each above table (i.e. area of each piece and number of pieces).
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