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Understanding Growth and Decay: Exponential Models and Applications

This section explores the concepts of exponential growth and decay using mathematical models. It outlines Theorem 5.16, which states that if y is a differentiable function of time t that remains positive, there exists a proportionality constant k. Positive k indicates growth, while negative k signifies decay. Through examples, such as the population of bacteria and the decay of radium, we derive formulas that help predict behavior over time, including calculations for future values and half-lives.

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Understanding Growth and Decay: Exponential Models and Applications

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  1. Section 5.6: Growth and Decay Model Theorem 5.16: If y is a differentiable function of t such that y > 0 and , for some constant k, then C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0.

  2. Example 1 When t = 0, y = 5, and when t = 3, y = 10. Given that the rate of change of y with respect to t is directly proportional y, find the value of y when t = 6.

  3. Example 2 Bacteria increase from 600 to 1800 in 2 hours. If the rate of increase is directly proportional to the number of bacteria, write a formula that will allow you to calculate the number of bacteria at the end of four hours, then five hours.

  4. Example 3 Radium decays exponentially and has a half life of 1600 years. Find a formula for the amount remaining after t years if you start with 50 mg. When will there be exactly 20 mg left?

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