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6.4 Exponential Growth and Decay

6.4 Exponential Growth and Decay. Glacier National Park, Montana Photo by Vickie Kelly, 2004. Greg Kelly, Hanford High School, Richland, Washington.

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6.4 Exponential Growth and Decay

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  1. 6.4 Exponential Growth and Decay Glacier National Park, Montana Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington

  2. The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep present (at least for awhile.) So does any population of living creatures. Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account. If the rate of change is proportional to the amount present, the change can be modeled by:

  3. Rate of change is proportional to the amount present. Divide both sides by y. Integrate both sides.

  4. Integrate both sides. Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication.

  5. Since is a constant, let . Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication.

  6. Exponential Change: If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay.

  7. How long would it take a $1200 investment to be worth $2000 if it was compounded continuously at 3.5% interest?

  8. How long would it take an investment to double if it was compounded continuously at 2.25% interest?

  9. Suppose that an initial population of 10,000 bacteria grows exponentially at a rate of 1% per hour. Find a formula that represents the number of bacteria present t hours later. How long would it take for the bacteria to reach 45,000?

  10. In the movie Pay It Forward each person was supposed to do a big favor for three more people. In turn, each of those people would do a favor for three more people, and so on. If you convinced 19 people to pay it forward on the first day and after 10 days, 193 people are involved, how many people will be involved 30 days after your “Pay It Forward” project began? Assume exponential growth is modeled during that time.

  11. Carbon-14 has a half-life of approximately 5700 years. What is the decay rate (k) of Carbon-14?

  12. In 1988, the Vatican authorized the British Museum to date a cloth relic known as the Shroud of Turin, possible the burial shroud of Jesus. This cloth contained the negative image of a human body, widely believed to be Jesus of Nazareth. The British Museum found the fibers in the cloth contained 92% of their original carbon-14. According to this information, what was the estimated age of the shroud?

  13. Suppose that the city of Newport had a population of 10,000 in 1987 and a population of 12,000 in 1997. Assuming an exponential growth model, in what year will the population reach 20,000?

  14. Homework Page 427 #21-23 Exponential Growth wkst

  15. Solutions to Page 427 #21-23 • 1,118,04 • 164,445,652 • a) • b) approx. 154,237.4 • c) 2.4 hours

  16. A question on the 2008 AP exam:

  17. A question on the 2008 AP exam:

  18. Day 2

  19. A certain isotope of sodium (Na-24) has a half-life of 15 hours. How long does it take for material to decay to 10% of its original amount?

  20. Half-life: Half-life

  21. If the growth rate (or decay rate) of a population, P, is proportional to the population itself, we say : In other words, the larger the population, the faster it grows. The smaller the population, the slower it grows. Solving this differential equation results in the population growth model :

  22. The population of gators in the Hillsborough river is growing at a rate proportional to the population. From a population of 50 on March 1st, the number of gators grows to 65 in 30 days. If the growth continues to follow the same model, how many days after March 1st will the population reach 100?

  23. The radioactive decay of a substance can be modeled by the differential equation where t is measured in years. Find the half-life of the substance. Round your answer to the nearest hundredth year.

  24. In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. In a certain reaction, the change of the substance satisfies the differential equation where y is measured in grams and t is measured in hours. If there are 100 grams of the substance when t = 0, how many grams will be left after the first hour?

  25. Suppose that electricity is draining from a capacitor at a rate proportional to the voltage V across its terminals and that, if t is measured in seconds, How long will it take the voltage to drop to 20% of its original value?

  26. A certain population is growing at a continuous rate so that the population doubles every 11 years. How long does it take for the population to triple?

  27. Homework

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