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\def\header{{\hfill Exam 4
\hfill WHITE
\hfill A. Miller
\hfill Fall 2005
\hfill Math 210\hfill}}

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\begin{document}

\bigskip

{ \bf Show all work.

Simplify your answer.

Circle your answer. }

\bigskip

No books, no calculators, no cell phones, no pagers, no
electronic devices of any kind.


\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\bigskip
Circle your Discussion Section:
\begin{verbatim}

   343  T   12:05--12:55        1412 STERLING
   344  R   12:05--12:55        1327 STERLING
   345  T   13:20--14:10        1327 STERLING
   346  R   13:20--14:10        1327 STERLING 

\end{verbatim}
\bigskip

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
   \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

\bigskip

Solutions will be posted shortly after the exam:
www.math.wisc.edu/$\sim$miller/m210

\setcounter{page}{0}

\newpage



\prob{10} % 1.3-19 
Let $A$ and $B$ be subsets of a universal set $U$.  Suppose
$n(U)=50$, $n(B)=30$ and $n(A\cup B^\prime)=30$.  Find
$n(A\cap B)$.

\answer 10

\prob{12} % 6.1
Let 
$$A=\left[
\begin{array}{rrr}
 2  &  -1  &  1 
\end{array}\right]
\;\;\;\;\;\;\;\mbox{ and } \;\;\;\;\;\;\;
B=\left[
\begin{array}{r}
 1  \\ 2 \\ -1   
\end{array}\right]$$

(a) Find $AB$ or state that it is an undefined product 

\bigskip
(b) Find $BA$ or state that it is an undefined product

\answer \par (a) $-1$ or $[-1]$
\par (b)
$$
\left[
\begin{array}{rrr}
 2  &  -1  &  1  \\
 4  &  -2  &  2  \\
 -2  &  1  &  -1 \\
\end{array}\right]
$$

 \prob{12} %  7.2-17
A set $F$ is described the inequalities
 $$ 3y+x\leq 14 $$
 $$ 3y-4x+26\geq 0$$
 $$ x\geq 2 $$
Graph the set $F$ and find the coordinates of the corner points. 

\answer $F$ is the interior and boundaries of the triangle
with corners $(2,-6),(8,2),(2,4)$.


 \prob{12} % 7.3
Jane's Juicy Fruit Stand sells two kinds of fruit drinks, 
Tangaid and Pungaid.  One glass of Tangaid contains 5 ounces
of orange juice and 3 ounces of lemon juice.  One glass of
Pungaid contains 2 ounces of orange juice and 6 ounces of lemon
juice.  Jane has available 200 ounces of orange juice and
360 ounces of lemon juice.  Her profit is 50 cents per glass
of Tangaid and 40 cents per glass of Pungaid.  How many glasses
of each kind should she produce to maximize her profit?   

\answer 20 Tangaid and 50 Pungaid.

 \prob{12} %  8.1-11
A Markov chain has the following transition matrix:
 $$
 \left[
 \begin{array}{rrr}
  .4  &  .1  &  .5 \\
  .1  &  .1  &  .8 \\
  .6  &  .2  &  .2 \\
 \end{array}\right]$$
On the first observation it is in state 1.

(a) What is the probability it is in state 1 on the second observation?

(b) What state is it most likely to occupy on the second observation?

\answer (a) $.4\;\;\;\;\;$ (b) state 3

 \prob{12} %  8.2-16
A not-so-enthusiastic student tends to miss class on Friday afternoons. She is
always in one of two states, absent A or present P, and transitions occur
according to the matrix:
 $$
  \begin{array}{r|rr}
       &   A  &   P \\
       \hline
   A   &  .3  &   .7 \\
   P   &  .6  &   .4  \\
 \end{array}$$
If the student is absent on a given Friday afternoon, find the probability she
will be present 2 weeks latter.

\answer $.49$

\finishtbl
\printanswers
\end{document}



