5.6 Applications and Models: Exponential Growth and Decay

Transcription

1 5.6 Applications and Models: Exponential Growth and Decay Many natural (and business/financial) processes build on themselves exponentially. We will see several examples. All of these examples are functions of t (time), measured from some starting time. If you start measuring a population in 1978, will correspond to 1978, and from then on will be the number of years after 1978 (so 1983 will be, 2000 will be, and t=16 corresponds to the year ). Standard Exponential Growth Formula: ( ) The variables represent different things in different contexts, but the general rules are: is time (usually in years, but occasionally the problem will say hours or days or something) ( ) is the amount of stuff (people, radioactive material, money, ) at time is the initial (original) amount of stuff (at ) is the exponential growth (or decay) rate, in decimal form: if the amount of stuff is growing, and if the amount of stuff is decreasing (decaying) Following are several different examples of exponential growth and decay; all use this same basic formula.

2 Population Growth A population growing with no resource/space restrictions grows exponentially, with standard exponential growth function as above. is time (usually measured in years for human populations, but days or hours for bacteria, etc.) P(t) is population at time t is initial population (when you start measuring it) is the exponential growth rate (if is negative, population is declining) In 2009, the population of Mexico was about million, and the exponential growth rate was 1.13% (so ). a) What is the exponential growth function? (The year corresponds to.) b) Estimate what the population will be in 2014 ( ) c) After how long will the population be double what it was in 2009?

3 5.6 continued Remember, the standard exponential growth/decay formula is ( ) Interest Compounded Continuously Instead of compounding times per year (as in the compound interest formula we discussed previously), you can compound interest so often it s essentially happening all the time. In this case, the formula for the money in the account is the standard exponential growth formula above. is time (years in account) ( ) is the amount of money you have in the account after years is the principal (P), the initial investment is the interest rate (as a decimal) Suppose that $5000 is invested at interest rate, compounded continuously, and grows to $ in 4 years. a) What is the interest rate? b) Find the exponential growth function. c) How much will be in the account after 10 years? d) After how long will the money have doubled?

4 Radioactive Decay A radioactive isotope decays into other isotopes or atoms over time, so there is less and less of the original isotope left as time goes on. At first (when there s more radioactive material present), the decay is fast, but then it slows down. The decay is modeled by the standard exponential growth/decay formula. is time ( ) is amount of radioactive material left at time is the initial amount of radioactive material present is the exponential decay rate. It must be negative when used in the formula, but usually people just talk about it as if it were positive: For example, we would say a decay rate is.0014 (or.14%), but in the formula we d use. Iodine-125 (commonly used as a marker in medical tests, and in radiation therapy for cancer) has a decay rate of 1.15% per day. (So will be measured in days, not years.) If.3 grams of I- 125 is placed in a tumor in a lab experiment, how long will it be until.03 grams is left?

5 5.6 continued Remember, the standard exponential growth/decay formula is ( ) Doubling Time/Half-life It is often convenient to talk about how long it takes an exponentially growing function to double in size, or an exponentially decaying function to decrease to half its original size. This does not depend on the amount you start with. Using the standard exponential growth/decay formula, we find the formula is the doubling-time or half-life is the exponential growth/decay rate. The absolute value bars around make it positive, so doubling-time/half-life will always be positive. How long will it take prices to double if inflation is 2.5% per year? Another The half-life of Carbon-14 (used in carbon-dating of ancient materials) is 5750 years. a) What is the decay rate for C-14? b) If a sample of C-14 is originally 100g and is measured again 2000 years from now, how much C-14 will be in the sample? c) If a sample of ancient mastodon bones contains only 22.8% of their original C-14, how old are the bones?

6 More Examples of Exponential Growth/Decay: 1. In 2009, the world population was 6.8 billion. The exponential growth rate was 1.13% per year. Estimate the population of the world in 2012 and 2020, and find when the world population will be 8 billion. 2. U.S. Exports increased from $12 billion in 1950 to $1550 billion (1.55 trillion) in Assuming the export growth rate is an exponential function, find the growth rate, and use it to estimate the value of exports in 1972 and The number of farms in the U.S. has decreased exponentially since In 1950 there were 5.6 million farms, and in 2008 that number was 2.2 million. At this decay rate, when will there be only 1 million farms in the U.S.? 4. The half-life of Carbon-14 is 5750 years. When the Dead Sea Scrolls were analyzed, it was found the linen wrapping them had lost about 22.3% of its original C-14. How old was the linen wrapping?