Difference between revisions of "Math 22 Exponential Growth and Decay"

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   is proportional to the quantity present at any time <math>t</math>, then  is of the form
 
   is proportional to the quantity present at any time <math>t</math>, then  is of the form
 
   <math>y=Ce^{kt}</math>
 
   <math>y=Ce^{kt}</math>
−
   where <math>C</math> is the initial value and <math>k</math> is the constant of proportionality. Exponential growth is indicated by <math>k>0</math> and exponential decay by <math>k<0</math>.
+
   where <math>C</math> is the initial value and <math>k</math> is the constant of proportionality.  
 +
  Exponential growth is indicated by <math>k>0</math> and exponential decay by <math>k<0</math>.
 +
==Guidelines for Modeling Exponential Growth and Decay==
 +
  1. Use the given information to write two sets of conditions involving <math>y</math> and <math>t</math>
 +
  2. Substitute the given conditions into the model <math>y=Ce^{kt}</math> and
 +
  use the results to solve for the constants <math>C</math> and <math>k</math>.
 +
  3. Use the model <math>y=Ce^{kt}</math> 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?
 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
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|-
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|At the initial time: <math>y_0=Ce^{kt_0}</math>, so <math>150=Ce^{kt_0}</math>
 +
|-
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|After 5 hours: <math>y_1=Ce^{k(t_0+5)}</math>, so <math>450=Ce^{k(t_0+5)}=Ce^{kt_0+5k}=Ce^{kt_0}e^{5k}</math>
 +
|-
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|Take the second equation and divided it by the first equation, we get:
 +
|-
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|<math>\frac{450}{150}=\frac{Ce^{kt_0}e^{5k}}{Ce^{k(t_0+5)}}</math>
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|-
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|<math>3=e^{5k}</math>
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|-
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|Now, consider the equation after 10 hours: <math>y_2=Ce^{k(t_0+10)}=Ce^{k[(t_0+5)+5]}=Ce^{k(t_0+5)}e^{kt}=450.3=1350</math>
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|}
 +
 
 +
'''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.
 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|Consider: <math>y=Ce^{kt}</math>
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|-
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|So, <math>20856=33000e^{k3}</math>, then <math>\frac{20856}{33000}=\frac{79}{125}=e^{3k}=(e^k)^3</math>
 +
|-
 +
|Then, <math>\sqrt[3]{\frac{79}{125}}=e^k</math>
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|-
 +
|Then, <math>\ln e^k=\ln {\sqrt[3]{\frac{79}{125}}}</math>
 +
|-
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|Therefore, <math>e=\ln {\sqrt[3]{\frac{79}{125}}}</math>
 +
|}
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 +
 
 +
 
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Latest revision as of 07:18, 12 August 2020

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:
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,


Return to Topics Page

This page were made by Tri Phan

Retrieved from "https://wiki.math.ucr.edu/index.php?title=Math_22_Exponential_Growth_and_Decay&oldid=2474"