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Growth

Exponential. Growth. and. growth or decay constant. time. amount at time t. amount initially. For k > 0, k is the growth constant or rate of growth. For k < 0, k is the decay constant or rate of decline. k > 0 growth curves. k < 0 decay curves. Radiocarbon Dating.

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Growth

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  1. Exponential Growth and growth or decay constant time amount at time t amount initially For k > 0, k is the growth constant or rate of growth For k < 0, k is the decay constant or rate of decline

  2. k > 0 growth curves k < 0 decay curves

  3. Radiocarbon Dating All living things contain Carbon 14. When they die, the C-14 begins to decay. We can determine how long something has been dead by the amount of C-14 left.

  4. In 1940 a group of boys walking in the woods near the village of Lascaux in France suddenly realized their dog was missing. They soon found him in a hole too deep for him to climb out. One of the boys was lowered into the hole to rescue the dog and stumbled upon one of the greatest archaeological discoveries of all time. What he discovered was a cave whose walls were covered with drawings of wild horses, cattle and a fierce-looking beast resembling a modern bull. In addition, the cave contained the charcoal remains of a small fire, and from these remains scientists were able to determine that the cave was occupied 15,000 years ago.

  5. Charcoal left from the logs contains Carbon-14

  6. By chemical analysis it has been determined that the amount of C-14 remaining in the samples of the Lascaux charcoal was 15% of the amount such trees would contain when living. The half-life of C-14 is approximately 5600 years. Solve this equation for k.First divide both sides by Ao Take the ln of both sides Divide by 5600 to find k The half-life of a radioactive element is the time it takes for 1/2 the initial amount to decay.

  7. By chemical analysis it has been determined that the amount of C-14 remaining in the samples of the Lascaux charcoal was 15% of the amount such trees would contain when living. The half-life of C-14 is approximately 5600 years. Take the ln of both sides Divide both sides by Ao Divide both sides by k So assuming the paintings were made at the time of the fire in the cave, they are approximately 15,000 years old.

  8. A culture of bacteria obeys the law of uninhibited growth. If 500 bacteria are present initially and there are 800 after 1 hour, how many will be present in the culture after 5 hours? How long is it until there are 20,000 bacteria? You’ll need to determine the k value by using info that tells you after a certain time it was a certain amount and subbing these values in for t and A. (1) 800 500 500 500 Notice the k is positive this time since this is exponential growth instead of decay. Now we know k we are ready to answer questions about the bacteria.

  9. 500 A culture of bacteria obeys the law of uninhibited growth. If 500 bacteria are present initially and there are 800 after 1 hour, how many will be present in the culture after 5 hours? How long is it until there are 20,000 bacteria? 20,000 500 500 (5) (5) 500 Solve for t Punch buttons in your calculator using the k value we already found.

  10. Another exponential equation that occurs in nature and involves the natural base e is called Newton’s Law of Cooling Temperature at time t Initial temperature time constant you determine knowing a time and temperature Surrounding Temperature

  11. 70 70 450 A pizza baked at 450F is removed from the oven at 5:00 pm into a room that is a 70F. After 5 minutes the pizza is at 300F. At what time can you begin eating the pizza if you want its temperature to be 135F? 70 70 450 We first find k by seeing that after 5 min. it was 300° 5 5 solve this for k

  12. A pizza baked at 450F is removed from the oven at 5:00 pm into a room that is a 70F. After 5 minutes the pizza is at 300F. At what time can you begin eating the pizza if you want its temperature to be 135F? Now that we have k we are ready to answer the question asked. So at 5:18 you can be munching away. 135 70 70 450 solve this for t

  13. One last type of equation we’ll look at is called the Logistic Growth Model which is limited growth instead of uninhibited growth. carrying capacity - the limit or number the population approaches as time approaches infinity population at time t Since this is a negative exponential, as t gets large it gets small. As t , this term  0 and the population  c.

  14. The logistic growth equation below relates the proportion of U.S. households that own a VCR to the year. Let t = 0 represent 1984, t = 1 represent 1985, and so on. 0.9 = 90% Determine the maximum proportion of households that will own a VCR. (Hint: this is the carrying capacity). What proportion of the U.S. households owned a VCR in 1984? (Hint: that is when t = 0)

  15. The logistic growth equation below relates the proportion of U.S. households that own a VCR to the year. Let t = 0 represent 1984, t = 1 represent 1985, and so on. need to solve for t by first multiplying both sides by the denominator 0.8 When will 0.8 (80%) of the households own a VCR? Isolate the exponential

  16. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

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