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MTH 112

MTH 112. Section 3.5 Exponential Growth & Decay Modeling Data. Overview. In this section we apply the concepts of exponential and logarithmic functions to population growth, half-life, and carbon dating. Population Growth. The exponential model

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MTH 112

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  1. MTH 112 Section 3.5 Exponential Growth & Decay Modeling Data

  2. Overview • In this section we apply the concepts of exponential and logarithmic functions to population growth, half-life, and carbon dating.

  3. Population Growth • The exponential model Describes the population, A, of a country t years after a starting year t0. A0 is the population in year t0, and k is the growth constant.

  4. Examples • The exponential model describes the population of a country, in millions, t years after 2003. Find the population of the country in 2003. • The exponential model describes the population of a country, in millions, t years after 2003. Use the model to determine when the population will be 718 million.

  5. More Examples • If the population of a country was 58.3 million in 2006, and is projected to be 54.4 million in 2030, find the growth (or decay) constant, k. • If the population of a county was 93.8 million in 2004 and the projected growth rate, k, is 0.00185, find the projected population in 2023.

  6. Carbon Dating • Very similar to population growth, except that k is negative. • When the amount of carbon-14 present is given as a percentage, then A/A0 is equal to that percentage (expressed as a decimal).

  7. Half-Life • The half-life of a radioactive element is the length of time required for that element to lose half of its radioactivity. • For example, if an element has a half-life of 20 years, then if it has 100 grams in 2014, in 2034 it will have 50 grams, in 2054 it will have 25 grams, in 2074 it will have 12.5 grams, and so on.

  8. More About Half-Life • When k is given, use the following formula to find t:

  9. More About Half-Life • When t is given, use the following formula to find k:

  10. Examples • A certain element has a decay rate, k, of 8.1% per year (k = -.081). Find the half-life of the element. • A certain element has a half-life of 3485 years. Find the decay rate, k. • Prehistoric cave paintings were discovered in a cave in France. The paint contained 8% of the original carbon-14. If k = -.000121, estimate the age of the paintings.

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