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\begin{titlepage}

\noindent
{\bf
\begin{tabular}{lll}
MA 162 & \hspace{2.8in} & Russell Brown \\
Final Exam & \hspace{2.8in} & 19 December, 1992
\end{tabular}}

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Answer all of the following questions.
Use the backs of the question papers for scratch paper.
Additional sheets are available if necessary.  No books
or notes may be used.  When answering these questions,
please be sure to 1) check answers when possible, 2)
clearly indicate your answer and the reasoning used to
arrive at that answer {\bf (unsupported answers will receive
NO credit)}, and 3) label all variables and equations.

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\begin{center}
\begin{tabular}{|r|c|r|}
\hline
Question & Score & Total \\
\hline \hline
1 && 15 \\
\hline
2 && 15 \\
\hline
3 && 15 \\
\hline
4 && 15\\
\hline
5 && 15 \\
\hline
6 && 15 \\
\hline
7 && 15 \\
\hline
8 && 15\\
\hline
9 && 15 \\
\hline
10 && 15 \\
\hline \hline
Total && 150 \\
\hline  
\end{tabular}
\end{center}

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\noindent
Name \underline{$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $}
\vspace{.1in}

\noindent
Section \underline{$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $}
\vspace{.1in}

\noindent
Seat number \underline{$ ~~~~~~~~~~~~~~~~~~~~ $}

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\end{titlepage}

\begin{enumerate}

\item
Consider the following information describing the supply
and demand of new and improved $ \beta $-widgets at two prices:
\vspace{.1in}

\[ \begin{array}{c|c|c} \mbox{Demand} & \mbox{Supply} & \mbox{Price} \\
\hline 22 & 13 & \mbox{\$5} \\
8 & 20 & \mbox{\$12} \end{array} \]

\noindent
a)  Find the supply and demand curves under the assumption that 
they are linear.

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b)  Find the equilibrium price.

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c)  Suppose the price is set at \$10.  Will there
be surplus or shortage?

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\newpage

\item
a)  Perform the following matrix operations,
if possible.  If on operation is not
allowed, explain why. 


i)  $ \left ( \begin{array}{c} 2 \\ 3 \end{array} \right ) + ( 2 ~~ 3 ) $
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ii)  $ \left ( \begin{array}{c} 2 \\ 3 \end{array} \right ) ~ ( 2 ~~ 3 ) $ 
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iii)  $ \left ( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right )
~~ \left ( \begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array} \right ) $.
\vfill

\newpage

\noindent
b)  Let
$ A = \left ( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\
0 & 1 & 1 \end{array} \right ) , ~~~ B = \left ( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 1 & 1 \\
0 & 0 & 1 \end{array} \right ) $ \\
and $ C = \left ( \begin{array}{rrr} 1 & -1 & 0 \\
0 & 1 & 0 \\
0 & -1 & 1 \end{array} \right ) $. 
\vspace{.2in}
Determine if $B$ or $C$ is the inverse of $A$.
Justify your answer.

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\item
Consider the system of equations
\[ \begin{array}{rrrr}
w + & x + & 2y + & z = 1 \\
w + & 2x + & y - & z = 0 \end{array} \]
Find two solutions to this system.

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\newpage

\item
Consider the tableau
\vspace{.2in}

\begin{center}
$ \begin{array}{ccccccc}
P & x & y & s_1 & s_2 & s_3 & \mbox{RHS} \\
\hline 1 & 0 & 0 & -1 & 0 & -3 & 5 \\
0 & 1 & 0 & 0 & 0 & 0 & 4 \\
0 & 0 & 1 & 2 & 0 & 1 & 3 \\
0 & 0 & 0 & 1 & 1 & 2 & 4 \end{array} $
\end{center}

\noindent
a) List the basic and nonbasic variables.
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\noindent 
b) Give the coordinates of the associated corner point.
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\noindent
c) Perform exactly one simplex pivot.
\vspace{1.4in}

\noindent
d) Is the new tableau optimal?
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\newpage

\item
Consider the linear program
\[ \begin{array}{lr}
\min \: 2x + y & \\ \\
~~~~ \mbox{subject to}: & \\
& x \leq 4 \\
& x + y \geq 4 \\
& x - y \geq 0 \end{array} \]

a)  Sketch the feasible set.
\vspace{.4in}

\begin{picture}(270,270)
\put(0, 5){\vector(1,0){270}}
\put(5,0){\vector(0,1){270}}
\multiput(30,3)(25,0){10}{\line (0,1){4}}
\multiput(3,30)(0,25){10}{\line(1,0){4}}
\put(10,260){\makebox(10,10)[lb]{$y$}}
\put(260,10){\makebox(10,10)[lb]{$x$}}
\end{picture}

\newpage

\noindent
b) Give the optimal feasible point
and the optimal objective function value.
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c) Draw the level line of the objective function
which passes through the optimal feasible point.
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\newpage

\item
A candy company packages bubble gum, Hershey's Kisses
and mints into mixed packages of candy.  They have 10 lbs
of each type of candy.  They sell two mixes.  Mix I
contains 2 oz.\ of bubble gum, 3 oz.\ of Hershey's kisses and 4 oz.\
of mints.  Mix II contains 4 oz.\ of bubble gum,
1 oz.\ of Hershey's kisses and 2 oz. of mints.  Mix I produces
a profit of \$0.50 per package and Mix II
produces a profit \$0.75 per packages.  Write
down a linear program whose solution will give the most
profitable way of packaging the available ingredients.

\newpage

\item
Use the Lindo printout to answer
the questions below.

\noindent
a)  What is the objective function?
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\noindent
b)  What is the optimal feasible point?
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\noindent
c)  Suppose that the resource in Row 2 is increased by 10 units.
What will be the new optimal objective function value?
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\noindent
d) Suppose that the coefficient of $x$
is changed from 3 to 4.  What will be the new optimal
objective function value?
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\noindent
e) Suppose that the coefficient of $y$ is changed
from 5 to 8.  What will the new optimal objective function
value be?
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\newpage

\item
a)  A child has 7 teddy bears and 4 stuffed dragons.
How many ways can he pick one of each to put in his bed?
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b)  How many ways can a family of five
stand in one row for a photograph?
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c)  How many different 3 member groups
can we form from a class of 31?
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\newpage

\item
On the third exam, 100 students took the exam,
82 came to class to collect their test paper and 82 passed the 
exam.  Seventy-six students passed the exam and
collected their test paper.

\noindent
a) How many students failed the exam and did not
collect their test?
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\noindent
b) Are the events $C$ of collecting the exam and $P$
of passing the exam independent?  Justify your answer.
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\newpage

\item
a) Suppose that you have 7 coins in your pocket.  Two
quarters, 2 nickels and 3 dimes.  What is the expected value of
picking a coin at random?  Assume that each coin is equally
likely to be drawn.
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\noindent
b) Bill makes 75\% of his free throws.  Assuming
that consecutive shots are independent, what is the probability of
Bill making exactly 3 shots in 4 attempts?
\vfill

\end{enumerate}
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