Math 22 Exponential Growth and Decay

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Exponential Growth and Decay

 If  is a positive quantity whose rate of change with respect to time
 is proportional to the quantity present at any time , then  is of the form
 
 where  is the initial value and  is the constant of proportionality. 
 Exponential growth is indicated by  and exponential decay by .

Guidelines for Modeling Exponential Growth and Decay

 1. Use the given information to write two sets of conditions involving  and 
 2. Substitute the given conditions into the model  and 
 use the results to solve for the constants  and .
 3. Use the model  to answer the question.

Exercises

1) The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 150 bacteria at a given time and 450 bacteria 5 hours later. How many bacteria will there be 10 hours after the initial time?

Solution:  
At the initial time: , so
After 5 hours: , so
Take the second equation and divided it by the first equation, we get:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{450}{150}=\frac{Ce^{kt_0}e^{5k}}{Ce^{k(t_0+5)}}}
Now, consider the equation after 10 hours:

2) A sports utility vehicle that costs $33000 new has a book value of $20856 after 3 years. Assume the value of the vehicle exponential decay over time. Find this exponential model.

Solution:  
Consider:
So, , then
Then,
Then,
Therefore,


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