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\large 
\underline
{MA2160 \hspace{5mm}Calculus II with Technology \hspace{50mm} Ch. 11-5}

Now that we can solve differential equations analytically, we can solve application problems that utilize differential equations


\Large
Goal
\large
\begin{enumerate}
\item Solve real world problems using differential equations
\end{enumerate}
There are numerous real world problems that can be solved by using differential equations.  Typical of these kinds of problems are exponential growth and decay problems and Newton's Law of cooling.
\begin{enumerate}


\item  Population growth: 
\begin{enumerate}
\item A population of bacteria continuously grows at a rate proportional to the current population.  If the population doubles in size every 12 hours, what is the proportional growth rate per hour?
\vfill
\item If the colony started with 1500 bacteria, at what time will it reach 1,000,000?
\vfill
\end{enumerate}  


\item  Exponential decay: 
\begin{enumerate}
\item A certain radioactive substance continuously loses 5\% of its radioactive material each day.  If after 30 days of decay there remains 45 $gm$ of radioactive material, how much radioactive material was there originally?
\vfill
\item How much more time must elapse for there to be only 10 $gm$ of radioactive material left?
\vfill
\end{enumerate}   

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\item  Newton's law of cooling:  Newton stated that a substance heats or cools proportionally to the difference of the temperature of the substance and the area surrounding it. 

\begin{enumerate}
\item If you put tap water (60 degrees) into a freezer that maintains a constant temperature of 20 degrees, and the water cools at a proportional rate of 10\% an hour, at what time will the water freeze (32 degrees)?
\vfill
\item If you put boiling water (212 degrees) in the same freezer, how long will it take to freeze?
\vfill

\end{enumerate} 
\end{enumerate}
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