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This section explores the concept of half-life using iodine-123 as an example, illustrating how to model the decay of a substance over time with a decreasing exponential function. Starting with 100 grams of iodine-123 and knowing its half-life is 13 hours, we derive an equation to find the remaining quantity after t hours and calculate the time needed for it to decay to 10 grams. Additionally, we discuss doubling time in a financial context, presenting a scenario of investing $1000 at an annual interest rate of 6.5% and determining when the investment will double.
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The half-life of a substance is the amount of time it takes for a decreasing exponential function to decay to half of its initial value • The half-life of iodine-123 is about 13 hours. You begin with 100 grams of iodine-123. • Write an equation that gives the amount of iodine remaining after t hours • Hint: You need to find your rate using the half-life information • Determine the number of hours for your sample to decay to 10 grams
Doubling time is the amount of time it takes for an increasing exponential function to grow to twice its previous level • Suppose we put $1000 in an account paying 6.5% compounded annually • Write an equation for the balance B after t years • When will the amount in our account double?
Any exponential function can be written as Q = abt or Q = aekt • Then b = ekt so k = lnb • Convert the function Q = 5(1.2)t into the form Q = aekt • What is the annual growth rate? • What is the continuous growth rate? • Convert the function Q = 10(0.81)t into the form Q = aekt • What is the annual decay rate? • What is the continuous decay rate? • In your groups try problems 7, 10, and 32
