exponential deacay
big idea - decay at a constant percentage rate can be described by an expression of the form bg^x, where 0 < g < 1 and the variable is in the exponet.
The growth factor and the type of exponential change
Three children were arguing. Tammy said, " If you multiply 5 by a positive numbe, the answer is always greater than 5." Nancy said, " No, your wronge! I can multiply 5 by something and get an answer less than 5." Leo said, "I acn buy somey=thing and get 5 for an answer."
In the childrens argument, what matters is how the multiplier compares to 1. If Nancy chooses a multipier between 0 and 1, multipling by 5 gives a result that is less than 5. For example, 5 x 1/10 = 1/2. Of course, Leon can do 5 x 1 and get 5 for answer.
This relates to exponential equations of the form y = b x g^x because the growth factor g can be greater than, equal to, or less than 1. In the last lesson, you saw only situation in which the growth factor was greater than 1, so there was an increase over time ibn each case. While the growth factor always has to be postive, it can be less than 1. When this is true, there is a decrease over time. this happens in situations of exponential decay.
In the childrens argument, what matters is how the multiplier compares to 1. If Nancy chooses a multipier between 0 and 1, multipling by 5 gives a result that is less than 5. For example, 5 x 1/10 = 1/2. Of course, Leon can do 5 x 1 and get 5 for answer.
This relates to exponential equations of the form y = b x g^x because the growth factor g can be greater than, equal to, or less than 1. In the last lesson, you saw only situation in which the growth factor was greater than 1, so there was an increase over time ibn each case. While the growth factor always has to be postive, it can be less than 1. When this is true, there is a decrease over time. this happens in situations of exponential decay.
Examples of exponential decay
Psychologists use exponential decay models to describe learning and memory loss. In Example 1, the growth factor is less than 1 so the amount remembered decreases.
example 1
Assume that each day after cramming, a student forgets, 20% of the vocabulary words learned the day before. A student crams for a French test on Friday by learning 100 vocabulary words Thursday night. But the test is delayed from Friday to Monday. If the student does not study over time weekends, how many words is he or she likely to remember on Monday.
Day Day Number Number of Words Remembered
Thursday 0 100
Friday 1 100(0.80) = 80
Saturday 2 100(0.80)(0.80) = 100(0.80)^2 = 64
Sunday 3 100(0.80)(0.80)(0.80)(0.80) = 51.2
Monday 4 100(0.80)(0.80)(0.80)(0.80) = 100(0.80)^4 = 40.96
Day Day Number Number of Words Remembered
Thursday 0 100
Friday 1 100(0.80) = 80
Saturday 2 100(0.80)(0.80) = 100(0.80)^2 = 64
Sunday 3 100(0.80)(0.80)(0.80)(0.80) = 51.2
Monday 4 100(0.80)(0.80)(0.80)(0.80) = 100(0.80)^4 = 40.96
graphs and growth factors
Exponential growth exponential decay are bothdescribed by an equation of the same form y = b x g^x.
vocabulary
Exponential Decay- A situation in which y = bg^x and 0 < g < 1
Half-Life- The time it takes for one half the amount of an element to decay.
Half-Life- The time it takes for one half the amount of an element to decay.