Math 22 Exponential Growth and Decay

Exponential Growth and Decay

 If $y$ is a positive quantity whose rate of change with respect to time
 is proportional to the quantity present at any time $t$, then  is of the form
 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=Ce^{kt}}

 where $C$ is the initial value and $k$ is the constant of proportionality. 
 Exponential growth is indicated by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k>0}
 and exponential decay by $k<0$.

Guidelines for Modeling Exponential Growth and Decay

 1. Use the given information to write two sets of conditions involving $y$ and $t$
 2. Substitute the given conditions into the model Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=Ce^{kt}}
 and 
 use the results to solve for the constants $C$ and $k$.
 3. Use the model Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=Ce^{kt}}
 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: $y_{0}=Ce^{kt_{0}}$, so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 150=Ce^{kt_{0}}}
After 5 hours: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{1}=Ce^{k(t_{0}+5)}} , so 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 450=Ce^{k(t_0+5)}=Ce^{kt_0+5k}=Ce^{kt_0}e^{5k}}
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)}}}
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 3=e^{5k}}
Now, consider the equation after 10 hours: 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 y_2=Ce^{k(t_0+10)}=Ce^{k[(t_0+5)+5]}=Ce^{k(t_0+5)}e^{kt}=450.3=1350}

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: 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 y=Ce^{kt}}
So, 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 20856=33000e^{k3}} , then 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{20856}{33000}=\frac{79}{125}=e^{3k}=(e^k)^3}
Then, 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 \sqrt[3]{\frac{79}{125}}=e^k}
Then, 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 \ln e^k=\ln {\sqrt[3]{\frac{79}{125}}}}
Therefore, 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 e=\ln {\sqrt[3]{\frac{79}{125}}}}

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