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How Do You Solve a Word Problem with Exponential Decay?

Andy originally bought a baseball card for $20 hoping that it would increase in value. Unfortunately, the value of the card has decreased by half each year. If this decrease in value continues, what will the value of Andy’s baseball card be after 4 years?

Summary

  1. Decreasing by a constant RATE means we have an exponential function
  2. Decreasing by a constant AMOUNT would mean we would have a linear function

Notes

    1. The formula for exponential decay is f(t) = a(1-r)t
    2. 'a' is the initial amount
    3. 'r' is the rate of decrease
    4. 't' is the time in years
    5. 'a' and 'r' must both be greater than 0
    1. The formula for exponential decay is f(t) = a(1-r)t
    2. 'a' is the initial amount, which is $20
    3. 'r' is the rate of decrease, which is 50%
    4. 't' is the time, which is 4 years
    1. To convert a percent to a decimal, drop the percent symbol and move the decimal point two places to the left
    2. So 50% becomes 0.5
    1. Plug in 20 for 'a', 0.5 for 'r', and 4 for 't' into f(t) = a(1-r)t
    2. This gives us f(4) = 20(1-0.5)4
    1. Use the order of operations to simplify the right hand side
    2. First simplify inside the parentheses
    3. Then evaluate the exponent
    4. Then multiply what's left
    5. Don't forget the dollar sign on your answer!