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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 120 A - Spring 2006 \hfil \pp
\hfil Mid-Term Exam Number One\hfil  \pp
\hfil April 20, 2006 \hfil \pp
\hfil Hints and Answers \hfil \pp


\begin{enumerate}

\item This problem can be solved using methods 
that were used to solve problem 6.4 of the text, to which it is 
identical except for the dimensions.

Version 1 (the pizza is 9 by 21 inches):

\[
A(x) = \left\{
\begin{array}{ll}
\frac{9}{30}x^2 & \mbox{if $0 \le x \le 15$,} \\
94.5 - \frac{9}{12}(x-21)^2 & \mbox{if $15 \le x \le 21$}
\end{array}
\right.
\]


Version 2 (the pizza is 7 by 21 inches):

\[
A(x) = \left\{
\begin{array}{ll}
\frac{7}{26}x^2 & \mbox{if $0 \le x \le 13$,} \\
73.5 - \frac{7}{16}(x-21)^2 & \mbox{if $13 \le x \le 21$}
\end{array}
\right.
\]



\item
After imposing a coordinate system, find the equation of the line
from Casserole to Needham.  Then find the equation of the line that
is perpendicular to this line, and passes through Lakewood.
Find the intersection of the two lines, and you'll have the point
at which Jessie is closest to Lakewood.  
Use the distance formula to find the distance from that point to Lakewood,
and that's the answer.

Version 1: 11.3417172... miles.

Version 2: 10.1353376... miles.





\item Find the equation of the sloping line.  Say it is $y=mx+b$.
Then the rectangle with width $x$ has height $y=mx+b$, and so has area
equal to 
\[
A = x(mx+b) = mx^2+bx
\]
By finding the vertex of this function of $x$, you can find the width
that achieves the maximum area, and the value of the maximum area.

Version 1: 52.25


Version 2: 55.25




\item After setting up a coordinate system (ideally with the center
of the forest at the origin, though other arrangements would also work),
find the equation of the line representing her path, and the equation
of the circle representing the boundary of the forest.
Find the intersections of the line and the circle.
This will tell you where she enters and where she exits the forest.
Use the distance formula to find the distance between these points
to get the distance she will walk through the forest.

Version 1: 9.5578 miles

Version 2: 7.6626 miles



\end{enumerate}




\end{document}


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