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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2018} \\
\textbf{Math 134 Midterm Exam 4} \\
\textbf{Friday, May 4}
\end{center}

\vskip 100pt

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 200pt


\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all work} on the test pages and place your final answer
in the box provided for the problem.  Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.

\vskip 30pt


\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 25\  \hfil \\
\hline
II  & \hfil/ 20\  \hfil \\
\hline
III  & \hfil/ 40\  \hfil \\
\hline\hline
IV  & \hfil/ 15\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfill\eject
\noindent
I. The time $t$ (in seconds) between two successive incoming calls at a 
very busy consumer service call center is a random variable with pdf
$$f(t) = \begin{cases}
         5 e^{-0.2t} & {\rm if}\ t \ge 0\\
              0 & {\rm if}\ t < 0
\end{cases}
$$
\begin{enumerate}
\item[A.] (5) Circle the integral that represents the probability that the time
between successive calls is \emph{at least 1} second:
$$\int_0^1 5 e^{-0.2 t}\ dt \qquad\qquad \int_1^\infty 5 e^{-0.2 t}\ dt$$ 
\item[B.] (10) Evaluate the integral you circled in part A.  
\vskip 2.5truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[C.] (10)  The median time between calls is the time $m$ for which 
$P(t \le m) = P(t \ge m) = 1/2$.  Determine the median time between calls at this center.
\vskip 2.5truein
\MyAnswer{median time $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
II.  Use the attached table of normal curve areas to answer the following questions. 
\begin{enumerate}
\item[A.]  (10) If $Z$ has a standard normal distribution, what is $P(-1.00 < Z < 1.34)$?
\vskip 2.0truein
\MyAnswer{Probability $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (10) Bottles of cream soda have a stated volume of 20 fluid ounces, but 
the actual volume of the contents is normally distributed with mean 20.2 fluid oz.  
and SD $= .4$ fluid oz.  What is the probability that a randomly selected bottle 
contains more than 20 fluid oz. of soda?
\vskip 3.0truein
\MyAnswer{Probability $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
III.  (Differential Equations)
\begin{enumerate}
\item[A.] (20)  Find the general solution of the differential equation 
$$\frac{dy}{dx} = (1 + y^2)x^3\ln(x)$$ 
\vglue 5.0truein
\MyAnswer{$y = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.] (5)  Newton's Law of Heating/Cooling says that if an object is placed into 
a surrounding medium of constant temperature $A$, then the rate of change of the object's
temperature is proportional to the difference between the object's temperature and the 
surrounding temperature.  Circle the differential equation that corresponds to this 
statement:
$$\frac{dy}{dt} = k(y - A) \qquad\qquad \frac{dy}{dt} = \frac{k}{y - A}$$
\end{enumerate}

\vfill\eject
\begin{enumerate}
\item[C.]  (15) A roasted chicken is taken out of the oven at a temperature of 
$190^\circ C$ and left to cool on a counter in a room maintained at a constant temperature
of $20^\circ$ C. 
After 10 minutes the temperature of the chicken has decreased to $100^\circ$ C. 
How long will it take for the temperature to reach $45^\circ$ C? 
\vglue 6.5truein
\MyAnswer{Time $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
IV.  
\begin{enumerate}
\item[A.]  (5)  What is the future value of $\$100,000$ at an interest rate of $5\%$, compounded
monthly for $15$ years?
\vskip 2.0truein
\MyAnswer{FV $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (5)  Suppose you make monthly payments of $ \$ 700$ for 15 years at $5\%$, what
is the future value of that stream of payments?
\vskip 2.0truein
\MyAnswer{FV $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[C.]  (5)  Suppose you were making those payments from part B to a bank in return for a loan 
of $\$100,000$.  At the end of the 15 years, would you still owe the bank, or would 
you have overpaid?  Explain.


\end{enumerate}
\end{document}

\vskip 200pt 
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\end{document}

