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MA162 \hfill Supplemental problems

\begin{enumerate}

%Below is the collection of all the supplemental problems I have ever
%assigned. They are unsorted and contain duplications.






\item
Suppose that the supply and demand curves for 162 textbooks
are 
\[ \begin{array}{l}
q = 10 p ~~~ \mbox{(supply)} \\ \\
q = 3000 - 50 p ~~~ \mbox{(demand)} \end{array} \]

a) Determine the equilibrium price and the quantity sold.

\vfill

b) If the price is set at \$40, determine if there 
will be a shortage or a surplus.  Give the quantity of
shortage or surplus.

\vfill

\newpage

\item
Suppose that the manufacture of one 162 textbook
requires 289 sheets of paper, 2 oz.\ of glue and 1.1 
ft$^2$ of of cover board.  The manufacture of one 123 text
requires 148 sheets of paper, 0.8 oz.\ of glue
and 1.5 ft$^2$ of cover board.  The manufacture of one
113 text requires 312 sheets of paper, 3.1 oz.\ of glue
and 1.9 ft$^2$ of cover board.  A publishing company has
50,000 sheets of paper, 300 oz.\ of glue
and 150 ft$^2$ of cover board.  Write down a system of
equation whose solution is the number of 162, 123
and 113 textbooks that can be made.  \underline{Do not
solve the system}.

\newpage

\item
Let $ A = \left ( \begin{array}{cc} 1 & 2 \\ 2 & -1 \end{array} \right ) , ~~~
B = \left ( \begin{array}{ccc} 1 & 2 & 3 \\ -3 & 2 & 1 
\end{array} \right ) $,  
and $ C = \left ( \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 1 
& 2 \end{array} \right ) $.  \\
Compute each of the following.
\vspace{.2in}

a) $ A^{ -1 } $

\vfill

b)  $ A \cdot B $

\vfill

c)  $ A \cdot C $

\vfill

d)  $ C + B $

\vfill

\newpage

\item
The system
\[ \begin{array}{rlll}
x && - z & = 1 \\
x & + y & + z & = 2 \\
3x & + y & - z & = 4 
\end{array} \]
has infinitely many solutions.
\vspace{.2in}

a)  Suppose that $ ( x,y,z ) $ is a solution.
If $z$ is arbitrary express $x$ and $y$ in terms of $z$.



b)  Suppose that $ z = 1 $.  Find the corresponding solution.





\item
A student has math and English to study.  Her total study
time is 8 hours.  In order to complete her English paper,
she needs a minimum of 2 hours.  Finally, she wishes to spend
at least twice as long on math as on English.
\vspace{.2in}

a)  Write a system of linear inequalities which summarize
the above information.

b)  Graph the system.

c)  Label all corner points of the feasible set and give
the coordinates of each corner point.






 
  





\item
Plot the following supply and demand data on one graph and join the
points with a smooth curve.  Use your {\em graph } to answer the
following questions.

\begin{center}
\begin{tabular}{r||r|r}
Price in \$ & Quantity supplied & Quantity demanded
\\
\hline
\$10 	& 3,000 & 8,000\\
\$12 & 5,000 & 7,000\\
\$15 & 6,000 & 5,500 \\
\$20 & 7,000 & 3,000\\
\end{tabular}
\end{center}
\begin{enumerate}
\item Do you think that either the supply curve or the demand curve is
given by a linear function?

\item Estimate the equilibrium price from your graph.
\item For which prices is there surplus?

\end{enumerate}


\item Find the solution using Gauss-Jordan elimination.

$$
\begin{array}{rrrcr}
x& & -z & = & -1 \\
x& +2y & +2z& = & 6 \\
x& -4y & +z& = & 1.
\end{array}
$$


\item \begin{enumerate} \item Write down  a system that has no
solutions.
        \item Consider the system 
\begin{eqnarray*} w+ x+ y + 2z &  =& 5\\
                  w + 2x -y +3z & = & 5
\end{eqnarray*}
	\begin{enumerate} \item Give all solutions to this system.
		\item Find two solutions to this system. 
	\end{enumerate}
      \end{enumerate}

\item Compute the following matrix operations, if possible.  If an
operation is imposssible, explain why.

$$
\left(\begin{array}{rr}
1 & 2\\3 & 4\end{array}\right)
\left(\begin{array}{r}1\\ 3
\end{array}\right),
\qquad 
\left(\begin{array}{r}1\\ 3
\end{array}\right)
\left(\begin{array}{rr}
1&2\\3&4\end{array}\right),
\qquad 
\left(\begin{array}{r}1\\ 3
\end{array}\right)
+
\left(\begin{array}{rr}
1&2\\3&4\end{array}\right)
$$



\item
Mailing a letter costs $x$ cents for the first ounce and $y$ cents for
each additional ounce. Suppose that a four ounce letter costs \$1.19 and
a six ounce letter costs \$1.77. 

\begin{enumerate}
\item 
Write a linear system whose solution is $x$ and $y$. 
\item 
Solve the system.
\end{enumerate}%end prob1

\item Find the solution by your favorite method.  

$$
\begin{array}{rrrcr}
x&+y  & +z & = & 6 \\
x& -y & +z& = & 2\\
x&+2y & -z& = & 2.
\end{array}
$$

\item Find the solution by your favorite method.  Use the back if
necessary.  
$$
\begin{array}{rrrcr}
x& & -z & = & -1 \\
x& +2y & +2z& = & 6 \\
x& -4y & +z& = & 1
\end{array}
$$

\item Let A = $ \left( \begin{array}{rrr} 1&1&1 \\
0&1&1\\
1&1&0
\end{array} \right)$.  Which of the following matrices is the inverse
of $A$?
$$
B= \left( \begin{array}{rrr} 1&-1&0\\ -1&1&1\\ 1&0&-1 
\end{array}\right) \mbox{\quad or \quad}
C= \left( \begin{array}{rrr}
1&-1&0\\ -1& 1& 1\\ 1&0&1
	  \end{array} \right)
$$


\item
Plot the following supply and demand data on one graph and join the
points with a smooth curve.  Use your {\em graph } to answer the
following questions.

\begin{center}
\begin{tabular}{r||r|r}
Price in \$ & Quantity supplied & Quantity demanded
\\
\hline
\$10 	& 3,000 & 8,000\\
\$12 & 5,000 & 7,000\\
\$15 & 6,000 & 5,500 \\
\$20 & 7,000 & 3,000\\
\end{tabular}
\end{center}
\begin{enumerate}
\item Do you think that either the supply curve or the demand curve is
given by a linear function?

\item Estimate the equilibrium price from your graph.
\item For which prices is there surplus?

\end{enumerate}

\item For each of the following matrix operations, either give the
result or explain why it is impossible.
$$
\left( \begin{array}{rr} 1 & 2 \\ 4 & -1 \end{array}\right)
\left( \begin{array}{r} 2 \\3 \end{array}\right), \quad
\left( \begin{array}{rr} 1 & 2 \\ 4 & -1 \end{array}\right) +
\left( \begin{array}{r} 2 \\3 \end{array}\right), \quad
\left ( \begin{array}{r} 2 \\-1 \end{array}\right),
\left( \begin{array}{rr} 2& 3\end{array}{rr}\right).
$$

\item Compute the following matrix operations, if possible.  If an
operation is imposssible, explain why.

$$
\left(\begin{array}{rr}
1 & 2\\3 & 4\end{array}\right)
\left(\begin{array}{r}1\\ 3
\end{array}\right),
\qquad 
\left(\begin{array}{r}1\\ 3
\end{array}\right)
\left(\begin{array}{rr}
1&2\\3&4\end{array}\right),
\qquad 
\left(\begin{array}{r}1\\ 3
\end{array}\right)
+
\left(\begin{array}{rr}
1&2\\3&4\end{array}\right)
$$
\end{enumerate}


\item
\begin{enumerate}
\item
 Find the inverse of the matrix 
$$
\left(\begin{array}{rrr}
1&1&0\\
1&2&0\\
1&1&1
      \end{array}
\right)
$$


\item
\begin{enumerate}
\item Find two numbers whose sum is 22 and whose difference is 9.  Use
a $2 \times 2$ linear system to find your solution.

\item  Write down a system of equations which has no solutions.
\end{enumerate}


\item At the Smith family's favorite fast food restaurant, a
hamburger contains 10 grams of fat, 70 grams of carbohydrates and 20 
grams of protein.   French fries contain 40 grams of fat, 50 grams of
carbohydrates and 5 grams of protein. Milk shakes contain 60 grams of fat, 5
grams of carbohydrates and 15 grams of protein.  Suppose that the Smiths
have 200 grams of fat, 600 grams of carbohydrates and 150 grams of protein
at lunch. Write down a system of
equations whose solutions are the number french fries, milk shakes and
hamburgers that the family had for dinner. Be sure to indicate what
each variable in your system stands for.  DO NOT solve the resulting
system of equations.  

 
\item Solve the system 

\begin{eqnarray*}
2x+y +z &=& 3\\
x+y + 2z &=& 5\\
x+y+z &=& 2.
\end{eqnarray*}

\item Find two solutions to the system
\begin{eqnarray*}
x+y+z &=&7\\
w+2x+ y \quad &=& 7.
\end{eqnarray*}
\item
\begin{enumerate}
\item Find the inverse of the matrix
$$
A= \left( \begin{array}{rrr}
1& 2& 0\\
1& 1& -1\\
1& 2& -1
	  \end{array}\right) .
$$
\item Use the inverse you found in part a) to solve the system
\begin{eqnarray*}
x+ 2y \quad &=& 3\\
x+y -z &=&1\\
x+2y -z &=& -1
\end{eqnarray*}
\end{enumerate}



\item Suppose that each ounce of food A has 2 grams of fat, 3 grams of
carbohydrate and 2 units of Vitamin Z.  Each ounce of food B has 3
grams of fat, 1 gram of carbohydrates and 3 units of vitamin Z.
Suppose that one needs to consume at least 6 grams of carbohydrates,
no more than 12 grams of fat and at least 3 units of vitamin Z.
\begin{enumerate}
\item Write down a system of linear inequalities which describe how
many ounces of each food one can eat while satisfying the given
requirements. 
\item Graph the feasible set for the system of inequalities in part a.
\item Find all the corners of the feasible set you drew in part b.

\end{enumerate}



\item
One ounce of carrots contains 2 units of vitamin A, 1 unit of vitamin
B and 2 units of vitamin C.  One ounce of peas contains 1 unit of
vitamin A, 3 units of vitamin B and 2 units of vitamin C.  Suppose a
dietitian wishes to serve a mixture of peas and carrots which contains
a minimum of
4 units of vitamin A, 3 units of vitamin B and 3 units of vitamin C.
\begin{enumerate}
\item Write a system of linear inequalities which summarizes the above
problem.
\item Graph the feasible set.
\end{enumerate}

\item Solve the linear program by the corner point method.
$$
\begin{array}{lrcl}
\mbox{\rm min}& x+2y &&\\
\mbox{\rm subject to } & x+y & \geq & 10\\
& 3x+y & \geq& 19\\
& x& \geq & 0\\
& y & \geq & 0
\end{array}
$$
\item Do the two problems on the Lindo instructions.

\item Consider the feasible set 

\begin{eqnarray*}
x+y & \leq & 10\\
2x+y& \leq &12\\
x &\geq  &0\\
y & \geq & 0.
\end{eqnarray*}


\begin{enumerate}
\item Sketch the feasible set.  
\item Give the augmented coordinates of each corner point of the
feasible set.
\item Give the remaining coordinates of the point where $x= 1$ and
$s_1 = 7$.  Is this point feasible?
\end{enumerate}


\item Solve the following linear program by the simplex method.

$$\begin{array}{l}
\mbox{max }  x+2y +z\\
\mbox{subject to } x  +y +z \leq 20\\
\qquad \qquad x+3y + 2z \leq 90
  \end{array}
$$

\item Consider the linear program
$$\begin{array}{lrcl}
\mbox{max} & x+y+z &&\\
\mbox{subject to} & x+y+2z& \leq & 10\\
& 2x+y +4z & \leq & 10\\
& x & \geq & 0\\
& y & \geq & 0\\
& z & \geq & 0 
\end{array}
$$
\begin{enumerate}
\item Use Lindo to find the dual prices for the first two constraints.
\item Suppose the right-hand side of the constraint $ x+y+2z\leq 10$
is decreased from 10 to 6.  Use the dual price to find the new
profit.
\item Use Lindo to check your answer to part b).
\end{enumerate}



\item \begin{enumerate}
\item On two different Venn diagrams, shade the sets $(A\cup B) \cap
C$ and $A \cup ( B \cap C)$.  In general, are these two sets the same?

\item Suppose that in a class of $60$, 40 students passed the first
exam, 35 students passed the second exam and 30 passed both exams.
How many passed neither exam?  How many passed at least one exam?

      \end{enumerate}
\item \begin{enumerate}\item Consider the letters A, B, C, and D.
	\begin{enumerate} \item List the permutations of these letters
taken two at a time.
\item List the combinations of these letters taken two at a time.
	\end{enumerate}
Hint:  You can use the formulas for $C(n,k)$ and $P(n,k)$ to see if
you have the right number of entries on your list.

  
\item Suppose that we pick a letter from the alphabet and then choose
a number from the set $\{ 1,2,3,4,5,6,7,8,9,10\}$.  How many different
ways can these choices be made?  Explain how you arrived at your
answer.

      \end{enumerate}

\item \begin{enumerate}\item Suppose that $P(A) =0.5$, $P(B) = 0.6$,
and $P(A\cap B) = 0.4$.  Compute $P(B')$, $P(A\cup B)$ and $P((A\cup
B)')$. 
\item Suppose that you roll two die. What is the probability that the
two die sum to 7?
      \end{enumerate}



\item At the first test, there were 107 students. Of these, 75 were
men and 32 were women.  There were 36 students wearing hats, but only
two women with hats.

\begin{enumerate}
\item Compute the probability that a student is wearing a hat. 
\item Compute the conditional probability of wearing a hat given that
a student is male. 
\item Compute the conditional probability of being male, given that a
student  is wearing a hat.

\item Are the events of wearing a hat and being male independent?
Why?

\end{enumerate}

\item
Suppose that you flip a coin and then draw a number. If the coin comes
up heads, then you draw from $\{1,2,3,4\}$ and if the coin is tails,
then you draw from $\{1,2,3,4,5,6\}$.  
\begin{enumerate}
\item What is the probability of drawing a six?
\item What is the probability of drawing a four?
\item If you draw a four, what is the probability that the coin was a
head?

\end{enumerate}

\item Suppose that a basketball player makes 90\% of his free
throws. 
\begin{enumerate}
\item What is the probability of making five out of ten free throws?

\item Suppose that he shoots 10 free throws and receives a \$100 for
each free throw that he makes over 7. (Thus, he receives nothing for 7
free throws, \$100 for 8, \$200 for 9  and \$300 for 10.)  What is the
expected value of this game?
\end{enumerate}


\item Solve by the corner point method.
$$
\begin{array}{lrcl}
\mbox {max} & x+2y &&\\
\mbox{subject to } & x+y &\geq &3 \\
& 2x+3y & \leq & 12\\
& x &\geq& 0 \\
& y & \geq&0
\end{array}
$$
\item Solve the two problems given in the Lindo handout.
\item Consider the feasible set 
\begin{eqnarray*}
x+y & \leq &5\\
2x+3y & \leq & 12\\
x& \geq & 0\\
y & \geq &0.
\end{eqnarray*}
Give all of the augmented coordinates for the points where (a) $ x= 2$
and $ y=3$, (b) $x=2$ and $s_1 = -2$, (c) $x=7$ and $y=5$.
Which of the above points are feasible? Why?



\item Consider the tableau:
$$
\begin{array}{rrrrrr|r}
x  & y & z   & s_1 & s_2 &s_3 &RHS\\ 
\hline
 1 & 2 & 0   & 1   & 0   & 1  & 10 \\
0  & 1 &  1  & 1  & 0    & 0  & 5  \\
0  & 1 & 0   & 1    & 1  & 1  & 5  \\
\end{array}
$$
\begin{enumerate}
\item Write the constraints as equations.
\item Give the basic and nonbasic variables.
\item Give the corner point associated to this tableau.
\item If $y=2$, $ s_1 =2$ and $s_ 3 = 0$, find the remaining augmented
coordinates.  Is this point feasible?
\item Pivot so that the variable $y$ becomes basic and the variable
$z$ becomes nonbasic. Find the new corner point.
\end{enumerate}



\item Consider the simplex tableau:
$$
\begin{array}{rrrrrr}
P& x& y & s_1 & s_2 & RHS\\
\hline
1&0&0& -1& 1&2\\
0&1&0&1&0&3\\
0&0&1&1&1&4
\end{array}
$$
\begin{enumerate}
\item Pivot so that the basic variables are $P$, $x$ and $s_1$.
\item The pivot you performed in part a) is not the pivot prescribed
by the simplex method.  Why not?  Why is the tableau you obtained in
part a) not allowed as a tableau in the simplex method?
\end{enumerate}
\item 
\begin{enumerate}
\item Solve the linear program by the graphical method.
$$\begin{array}{lrcl}
\mbox{max} & x+y &&\\
\mbox{subject to} & x+2y &\leq & 4\\
& 2x+y & \leq & 5\\
& x & \geq& 0 \\
& y & \geq & 0
  \end{array}
$$
\item Solve the linear program by the graphical method.
$$\begin{array}{lrcl}
\mbox{max} & x+y &&\\
\mbox{subject to} & x+2y &\leq & 4\\
& 2x+y & \leq & 6\\
& x & \geq& 0 \\
& y & \geq & 0
  \end{array}
$$
\item Determine the dual price of the second constraint in the linear
program in part a).
\end{enumerate}


\item
\begin{enumerate}\item Shade the sets $(A\cup B) '$ and $A'\cap B'$ on two Venn
diagrams.  Are these two sets equal?

\item How many ways can you rearrange the letters in \lq LEWIS' if
repetitions are not allowed?

\item How many five letter words can you make from the  letters in `CLARK' if
repetitions are allowed?  (Hint: Parts b) and c) can be answered using
the multiplication principle.)
\end{enumerate}

\item Problem 12 is cancelled.
\item 
\begin{enumerate}
\item Compute $C(5,2)$, $P(5,2)$ and $C(12,10)$.  Your solution should
indicate how you carried out the arithmetic.  In particular, I do not
want you to use a calculator for these computations.

\item How many ways can you rearrange the letters in `RUSSELLBROWN'
using each letter only as often as it appears in Russell Brown's name?
(For this part you will need a calculator.)
\end{enumerate}


\item Let $A$ and $B$ be two events.  Suppose that $P(A)= 0.3$, $P(B)
=0.4$ and $P(A\cap B)= 0.1$.
\begin{enumerate}
\item Compute $P(A\cup B)$.
\item Compute $P( A'\cap B')$.
\end{enumerate}

\item (The due date on problem  15 has been changed from 11/15 to 11/17).
A sample space contains 80 outcomes.  Assume that all outcomes
are equally likely.  Let $X$ and $Y$ be two events in $S$ and suppose
that $n(X) = 50$,  $n(Y)=30$ and $n( (X\cup Y)') =20$.  
\begin{enumerate}
\item Draw a Venn diagram summarizing this information.
\item Are the events $X$ and $Y$ independent?
\item Are the events $X$ and $X'$ independent?
\end{enumerate}


\item Suppose that a pollster asks each respondent to roll a die and
then answer a yes or no question. If the die is a one or two, then the
person is answer yes if the last digit of their drivers license is odd
and no otherwise.  If the die is three through six, then the person is
to answer the question ``Do you like finite math?''.  Suppose that
40\% of the people asked respond yes, what is the probability that one
likes finite math.


\item An insurance company has found that a certain policy produces a
claim of \$500 with probability of 0.05 and a claim of \$10,000 with
probability of 0.001.  
\begin{enumerate}
\item What should they charge for this policy in order to make a
profit of \$10 per policy?

\item Which claims, on average, cost the company more:  the small
claims of \$500 or the large claims of \$10,000?

\end{enumerate}
\item A company has developed a test for disease $Y$.  Disease $Y$
occurs in 70\% of the patients to be tested.  The test is accurrate in
80\% of people with disease $Y$ and in 75\% of the people without
disease $Y$.  
\begin{enumerate}
\item Draw a probability tree which summarizes the above information.
\item If a person tests positive for disease $Y$, what is the
probability that the person is ill with disease $Y$?
\end{enumerate}



\item
\begin{enumerate}
\item 
An individual has \$500 to invest in stocks and bonds.  The stock
investment earns 10\% and the bond investment earns 6\%.  If his total
earnings are \$42, determine how he divided his money between stocks
and bonds.
\item 
Give an example of a $2\times 2$ system of equations with infinitely many
solutions.
\end{enumerate}%end prob1

\item Find the solution by your favorite method.  

$$
\begin{array}{rrrcr}
x& & -z & = & -1 \\
x& +2y & +2z& = & 6 \\
x& -4y & +z& = & 1.
\end{array}
$$
\item
\begin{enumerate}
\item A company manufacturers textbooks from cover board, paper and
glue.
One MA109 textbook uses 1.5 square feet of cover board, 135 sheets of
paper and 1.5 ounces of glue.  One MA123 textbook uses 2 square feet
of cover board, 120 sheets of paper and 1 ounce of glue.  An MA162
text book uses 2 square feet of cover board, 190 sheets of paper and
1.25 ounces of glue.  Suppose that the company has 200 square yards of
cover board, 20 reams of paper and 10 pints of glue.  Write a system
of equations whose solution tells how many of each book they should
make in order to use all available materials.

\item Find the solution to the system
\begin{eqnarray*}
w+ x+ 2y +z & = & 1\\
2w -x +3y + 2z & = & -1
\end{eqnarray*}
when $ x=1 $ and $ z= -1$.
\end{enumerate}

\item \begin{enumerate}
\item Let $ A = \left( \begin{array}{rr} a & b \\ c& d\end{array}\right)$
and suppose that $A^{-1} = \left(\begin{array}{rr}1 & 2\\ 1&
-1\end{array}\right)$. Solve for $x $ and $y$.
\begin{eqnarray*}
ax +by & = & 1\\
cx + dy &=& -3
\end{eqnarray*}

\item Find the inverse of
$$
\left(
\begin{array}{rrr}
1 & 0 & -2\\
2 & 1 & 0\\
-4 & -2 & 1
\end{array}
\right)
$$
\end{enumerate}


\item Suppose that each ounce of food A has 2 grams of fat, 3 grams of
carbohydrate and 2 units of Vitamin Z.  Each ounce of food B has 3
grams of fat, 1 gram of carbohydrates and 3 units of vitamin Z.
Suppose that one needs to consume at least 6 grams of carbohydrates,
no more than 12 grams of fat and at least 3 units of vitamin Z.
\begin{enumerate}
\item Write down a system of linear inequalities which describe how
many ounces of each food one can eat while satisfying the given
requirements. 
\item Graph the feasible set for the system of inequalities in part a.
\item Find all the corners of the feasible set you drew in part b.

\end{enumerate}


\item Suppose that each ounce of food A has 2 grams of fat, 3 grams of
carbohydrate and 2 units of Vitamin Z.  Each ounce of food B has 3
grams of fat, 1 gram of carbohydrates and 3 units of vitamin Z.
Suppose that one needs to consume at least 6 grams of carbohydrates,
no more than 12 grams of fat and at least 3 units of vitamin Z.
\begin{enumerate}
\item Write down a system of linear inequalities which describe how
many ounces of each food one can eat while satisfying the given
requirements. 
\item Graph the feasible set for the system of inequalities in part a.
\item Find all the corners of the feasible set you drew in part b.

\end{enumerate}


\item Solve the linear program by the corner point method.
$$
\begin{array}{lrcl}
\mbox{\rm min}& x+2y &&\\
\mbox{\rm subject to } & x+y & \geq & 10\\
& 2x+y & \geq& 14\\
& x& \geq & 0\\
& y & \geq & 0
\end{array}
$$
\item Do the two problems on the Lindo instructions.

\item Consider the feasible set 
\begin{eqnarray*}
x+y & \leq &5\\
2x+3y & \leq & 12\\
x& \geq & 0\\
y & \geq &0.
\end{eqnarray*}
Give all of the augmented coordinates for the points where (a) $ x= 2$
and $ y=3$, (b) $x=2$ and $s_1 = -2$, (c) $x=7$ and $y=5$.
Which of the above points are feasible? Why?
\item Solve the linear program by the simplex method.
$$
\begin{array}{lrcl}
\mbox{max } & x+y+2z&&\\
\mbox{subject to} & x+y+z &\leq  & 4\\
& 3x + z & \leq & 4\\
& x & \geq &0\\
& y & \geq & 0\\
& z & \geq & 0
\end{array}
$$

\item Consider the linear program
$$\begin{array}{lrcl}
\mbox{max} & x+y+z &&\\
\mbox{subject to} & x+y+2z& \leq & 10\\
& 2x+y +4z & \leq & 10\\
& x & \geq & 0\\
& y & \geq & 0\\
& z & \geq & 0 
\end{array}
$$
\begin{enumerate}


\item   Use the Lindo printout below to answer the following
questions.  You should consider each question separately. That is
after each question, assume that all coefficients and righthand sides
are returned to their original values.
\begin{enumerate}
\item Suppose that righthand side of the constraint in row 2 is
increased by 5.  What is the new optimal objective function value?
% allowed

\item Suppose that the righthand side of the constraint in row 3
changed to 15.  What is the new optimal objective function value?

\item Suppose the coefficient of $x$ increases to 5. What is the new
optimal objective function value?

\item Suppose the coefficient of y is increased by 4. What is the new
optimal objective function value?

\item Suppose that we must choose between increasing the righthand
side of row 2 by 3 or increasing the righthand side of row 3 by
3. Which increase  will  lead to a greater increase in the objective
function value? 

\begin{verbatim}


  MAX     4 X + 5 Y
  SUBJECT TO
         2)   2 X + 3 Y <=   15
         3)   4 X + 3 Y <=   20
  END

 LP OPTIMUM FOUND AT STEP      2

        OBJECTIVE FUNCTION VALUE

        1)      26.66667    

  VARIABLE        VALUE          REDUCED COST
         X         2.500000           .000000
         Y         3.333333           .000000


       ROW   SLACK OR SURPLUS     DUAL PRICES
        2)          .000000          1.333333
        3)          .000000           .333333

 NO. ITERATIONS=       2

 RANGES IN WHICH THE BASIS IS UNCHANGED:
 
                           OBJ COEFFICIENT RANGES
 VARIABLE         CURRENT        ALLOWABLE        ALLOWABLE
                   COEF          INCREASE         DECREASE
        X        4.000000         2.666667          .666667
        Y        5.000000         1.000000         2.000000
 
                           RIGHTHAND SIDE RANGES
      ROW         CURRENT        ALLOWABLE        ALLOWABLE
                    RHS          INCREASE         DECREASE
        2       15.000000         5.000000         5.000000
        3       20.000000         9.999999         5.000000
 
\end{verbatim}


\end{enumerate}




\item Use Lindo to find the dual prices for the first two constraints.
\item Suppose the right-hand side of the constraint $ x+y+2z\leq 10$
is decreased from 10 to 5.  Use the dual price to find the new
profit.
\item Use Lindo to check your answer to part b).
\end{enumerate}



\item Consider the linear program
$$\begin{array}{lrcl}
\mbox{max} & x+y+z &&\\
\mbox{subject to} & x+y+2z& \leq & 10\\
& 2x+y +4z & \leq & 10\\
& x & \geq & 0\\
& y & \geq & 0\\
& z & \geq & 0 
\end{array}
$$
\begin{enumerate}
\item Use Lindo to find the dual prices for the first two constraints.
\item Suppose the right-hand side of the constraint $ x+y+2z\leq 10$
is decreased from 10 to 5.  Use the dual price to find the new
profit.
\item Use Lindo to check your answer to part b).
\end{enumerate}



\item   Use the Lindo printout below to answer the following
questions.  You should consider each question separately. That is
after each question, assume that all coefficients and righthand sides
are returned to their original values.
\begin{enumerate}
\item Suppose that righthand side of the constraint in row 2 is
increased by 5.  What is the new optimal objective function value?
% allowed

\item Suppose that the righthand side of the constraint in row 3
changed to 15.  What is the new optimal objective function value?

\item Suppose the coefficient of $x$ increases to 5. What is the new
optimal objective function value?

\item Suppose the coefficient of y is increased by 4. What is the new
optimal objective function value?

\item Suppose that we must choose between increasing the righthand
side of row 2 by 3 or increasing the righthand side of row 3 by
3. Which increase  will  lead to a greater increase in the objective
function value? 

\begin{verbatim}


  MAX     4 X + 5 Y
  SUBJECT TO
         2)   2 X + 3 Y <=   15
         3)   4 X + 3 Y <=   20
  END

 LP OPTIMUM FOUND AT STEP      2

        OBJECTIVE FUNCTION VALUE

        1)      26.66667    

  VARIABLE        VALUE          REDUCED COST
         X         2.500000           .000000
         Y         3.333333           .000000


       ROW   SLACK OR SURPLUS     DUAL PRICES
        2)          .000000          1.333333
        3)          .000000           .333333

 NO. ITERATIONS=       2

 RANGES IN WHICH THE BASIS IS UNCHANGED:
 
                           OBJ COEFFICIENT RANGES
 VARIABLE         CURRENT        ALLOWABLE        ALLOWABLE
                   COEF          INCREASE         DECREASE
        X        4.000000         2.666667          .666667
        Y        5.000000         1.000000         2.000000
 
                           RIGHTHAND SIDE RANGES
      ROW         CURRENT        ALLOWABLE        ALLOWABLE
                    RHS          INCREASE         DECREASE
        2       15.000000         5.000000         5.000000
        3       20.000000         9.999999         5.000000
 
\end{verbatim}

 
\item \begin{enumerate}
\item On separate Venn diagrams, shade the sets $ (A\cup B) \cap C'$
and $(A'\cap B') \cup C$.
\item Suppose that $A $ and $B$ are subsets of a universe $U$. If
$n(U) = 50$, $n(A) = 20$, $n(B) = 40$ and $ A'\cap B'  = \emptyset$,
then compute $n(A\cap B)$.

      \end{enumerate}

\item \begin{enumerate}
\item Compute $C(12, 4)$ and $ P(12,4)$.
\item Consider the set of letters $\{A,B,C,D\}$. 
\begin{enumerate}
\item List all the
combinations of these letters taken three at a time.  
\item List all of the permutations of these letters taken two at a
time.

\end{enumerate}
\end{enumerate}




\item 
\begin{enumerate}
\item Draw the sets $ A \cap ( B\cup C)$ and $( A \cap B) \cup C$  on
separate Venn diagrams. In general, are these sets equal?
\item  Suppose that in a class, 40 students understand the simplex
method, 50 students understand the corner point method and 30 students
understand both.  If every student understands either the simplex
method or  the corner point method, how many students are in the
class?  Use a Venn diagram to illustrate your answer.
\end{enumerate}
\item 
\begin{enumerate}
\item How many permutations are there of the letters in SIX? List
them.
\item How many combinations are there  of the letters in HARE taken two at a
time? List them. 
\item How many combinations are there of the letters in HARE taken
three at a time? List them.
\end{enumerate}


\item Suppose that we flip 4 coins and think of the outcomes as 4
letter words using the letters H and T.  
\begin{enumerate}
\item How many outcomes are possible?
\item How many outcomes are all heads?
\item How many outcomes have at least one tail? Why?
\item How many outcomes have exactly two tails?
List them.
\item How many outcomes have exactly three tails?  List them.
\end{enumerate}


\end{enumerate}
\end{enumerate}
\item Suppose that you roll two dice. 
\begin{enumerate}
\item List the sample space for this experiment.
\item Compute the probability of obtaining at least one 6.
\item Compute the probability of obtaining exactly one 6.
\item Compute the probability of obtaining no sixes.
\item Compute the probability that the numbers on the two dice sum to 4.
\end{enumerate}



\item   \begin{enumerate} \item On two separate Venn diagrams, shade $
A' \cap B$ and $ (A\cup B )'$.  Are these sets the same?
\item If $A = \{ 1, 2, 4, 8\}$ and $B = \{1, 3, 5, 7\}$, list the
elements of $A\cup B$ and $ A \cap B$.

	\end{enumerate}


\item \begin{enumerate}
\item List all the combinations with two elements chosen from the
letters in  QUIT.
\item List all the permutations with two elements chosen from the
letters in QUIT.
\item Compute $P(4,2)$ and $C(4,2)$ and check to see if your lists in
parts a and b have the correct number of elements.
      \end{enumerate}



\item   Suppose that $A$ and $B$ are events with $P(A)= P(B) = 0.5$
and $P(A\cup B) = 0.8$.  
\begin{enumerate}
\item What is $P(A \cap B)$?
\item What is $P(A')$?
\item Are the events $A$ and $B$ independent?  As always, you should
explain how you arrived at your answer.

\end{enumerate}

\item Suppose that you flip 4 coins and will receive a payment of \$2
multiplied by the number of heads you obtain.

\begin{enumerate}
\item Compute the probabilities of obtaining 0, 1, 2, 3 or 4 heads.  
\item Why should your answers in a) sum to one?
\item What is the expected value of playing this game?
\end{enumerate}


\item  A company has developed a test for disease $Y$.  Disease $Y$
occurs in 70\% of the patients to be tested.  The test is accurate in
85\% of the people with disease $Y$ and 70\% of the people without
disease $Y$.

\begin{enumerate}

\item Draw a probability tree which summarizes the above information.
\item If a person tests positive for disease $Y$, what is the
probability that the person is ill with disease $Y$?

\end{enumerate}


\item 
An individual has \$600  invested  in stocks and bonds.  The stock
investment earns 10\% and the bond investment earns 6\%.  If his total
earnings are \$44, determine how he divided his money between stocks
and bonds.





\item 
Give an example of a $2\times 2$ system of equations with infinitely many
solutions.




\item At the Smith family's favorite fast food restaurant, a
hamburger contains 10 grams of fat, 70 grams of carbohydrates and 20 
grams of protein.   French fries contain 40 grams of fat, 50 grams of
carbohydrates and 5 grams of protein. Milk shakes contain 60 grams of fat, 5
grams of carbohydrates and 15 grams of protein.  Suppose that the Smiths
have 200 grams of fat, 600 grams of carbohydrates and 150 grams of protein
at lunch. Write down a system of
equations whose solutions are the number french fries, milk shakes and
hamburgers that the family had for dinner. Be sure to indicate what
each variable in your system stands for.  DO NOT solve the resulting
system of equations.  

 



\item
\begin{enumerate}\item Solve the system of equations
$$\begin{array}{rrrl}
x&+y& +z& =6\\
 & y& +z& =5\\
x& +y& & =3
\end{array}
$$

\item Solve the system of equations:
$$
\begin{array}{rrrl}
x&+y& +z& =4\\
 & y& +z& =1\\
x& +y& & =5
\end{array}
$$
%\vspace {1in}
     \end{enumerate}










\begin{enumerate}
\item
Find two solutions to the system

\begin{eqnarray*}
w+x+2y+z&=&4\\
w+2x-y+z &=& 4.
\end{eqnarray*}


\end{enumerate}


\begin{enumerate}
\item
 Find the inverse of the matrix 
$$
\left(\begin{array}{rrr}
1&1&0\\
1&2&0\\
1&1&1
      \end{array}
\right)
$$

\item Use the inverse in part a) to solve the system

\begin{eqnarray*}
x+y &=& 1\\
x+2y &=& 2\\
x+y+z &=& 3
\end{eqnarray*}
\end{enumerate}






\item
Use LINDO to find the solutions to the following problems.  Turn in
your LINDO printout along with your solutions.
\begin{enumerate}
\item You want to start a business which will make tables, chairs and
desks.  You have only 100 units of wood available and you know that it
takes 6 units of wood to make a table, 4 units to make a chair and 5
units of wood to make a desk.  You also have the constraint that you
only have 600 hours of labor available to you.  It takes 10 hours to
make a table, 15 hours to make a chair and 20 hours to make a desk.  

If you can earn a profit of \$60 per table, \$50 per chair and \$40
per desk, how many of each should you produce to maximize your
profit?  How much profit will you make?

\item We change our process so that now a chair will require 5 units
of wood, but the desks will only take 2 units of wood.  This new type
of chair will now take 35 hours to build.  All else remains the same.

How many of each product should we make now?  What is our profit?
\end{enumerate}




\item \begin{enumerate}
\item Graph the feasible set:
$\begin{array}[t]{l}
x+y \leq 9\\
x \geq 2y\\
x\geq 0, \  y \geq 0
\end{array}$



\item Consider the tableau:
$$
\begin{array}{rrrrrr|r}
x  & y & z    & s_1 & s_2 &s_3 &RHS\\ 
\hline
 1 & 2  & 0   & 1 & 0   & 1 & 10 \\
 0 &  1 &  1  & 1 & 0   & 0 & 5  \\
0  & 1  & 0   & 1 & 1   & 1 & 5  \\
\end{array}
$$
\begin{enumerate}
\item Write the constraints as equations.
\item Give the basic and nonbasic variables.
\item Give the corner point associated to this tableau.
\item If $x=2$, $ z =2$ and $s_2= 0$, find the remaining augmented
coordinates.  Is this point feasible?
\item Pivot so that the variable $y$ becomes basic and the variable
$z$ becomes nonbasic. Find the new corner point.
\end{enumerate}

\item  Consider the feasible set
\begin{eqnarray*}
x+y+z & \leq & 10\\
x  & \leq & 5\\
2x+y+z& \leq & 12\\
x\geq 0, y \geq 0, z \geq 0 & & 
\end{eqnarray*}
\begin{enumerate}
\item Introduce the augmented coordinates for this feasible set and
give the equations relating the decision variables $x$, $y$ and $z$ to
the slack variables $s_1$, $s_2$ and $s_3$.  

\item Suppose $x= 0$, $y=0$ and $z=0$.  Give the values of the slack
variables. 
\item Suppose$x= 4$, $y = 5$ and $z=6$.  Give the values of the slack
variables.  Is this point feasible?  As always, you should justify
your answer.

\item Suppose the slack variable for the first constraint is 10 and
$x=5$ and $y=5$.  Find the values of the remaining augmented
coordinates.  Is this point feasible? Why?
\end{enumerate}


\item Solve the linear program: \quad$ \begin{array}[t]{l}
\mbox{max } 3x+y\\
\mbox{Subject to}\\
\quad x+y\leq 9\\
\quad x\geq 2y\\
\quad x\geq 0, \  y \geq 0
				\end{array}$
\end{enumerate}


\item A mining company has two mines whose production is given the
table below.

\begin{center}
\begin{tabular}{|r|rrr|}
\hline
 & Lead & Tin & Platinum\\ \hline\hline
Mine 1 & 6 lbs/day& 3 lbs/day & 1 lb/day\\
Mine 2 & 7 lbs/day & 4 lbs/day & 2 lbs/day\\
\hline
\end{tabular}
\end{center}

Mine 1 costs \$20,000 a day to operate while Mine 2 costs \$15,000 per
day.  In order meet their orders they need to produce at least 700 lbs
of lead, 900 lbs of tin and 150 lbs of platinum.  Write down a linear
program whose solution gives the number of days the company should
operate each mine while minimizing its costs. Do not solve the linear
program.

\item Consider the following simplex tableau:
$$
\begin{array}{r|rrrrrrr|r}
Row & P&x_1&x_2&x_3&s_1&s_2&s_3&RHS\\ \hline
0   & 1& 0 & -1& -2& 0 &0  & 0 &8 \\
1   & 0&1  &  2 & 2&  0& 0 & 2 & 8\\
2   & 0 & 0&1  &1   & 0& 1 & 1 & 2 \\
3 &0 & 0&   3&  2   & 1&  0&  7& 6
\end{array}
$$
\begin{enumerate}
\item List the basic variables\rule{1in}{0.01in} \\[0.1in]
\item List the nonbasic variables \rule{1in}{0.01in}\\[0.1in]
\item Give the augmented coordinates of the corner associated to this \\[0.2in]
tableau.   \rule{1in}{0.01in}\\[0.1in]
\item Give the variable at the top of  the pivot column.
\rule{1in}{0.01in}
\\[0.1in]
\item Give the number of the  pivot row. \rule {1 in}{0.01in}
\\[0.1in]
\item Give the current value of the objective function.  \rule {1in}
{0.01in}
\\[0.1in]
\item Is this value the optimal value? Why or why not?
\end{enumerate}

\item Consider the tableau:
$$
\begin{array}{rrrrr|r}
P& x_1&x_2&s_1&s_2&RHS\\
\hline
1 &-1&1&0&0&2\\
0&1&2&0&1&2\\
0&1&3&1&0&1
\end{array}
$$
\begin{enumerate}
\item According to the simplex method, which pivot should you perform.
(That is, what will the new basic variables be after you pivot?)
\item Pivot so that $x_1 $ and $s_1 $ are the new basic variables.
(Note that this may not be the pivot prescribed by the simplex
method.)
\item Does the new tableau correspond to a corner point of the
feasible set?  Why or why not?
\end{enumerate}

\item How many three letter (sensible or nonsensible) words are there
which are of the form consonant, vowel, consonant.  Please explain your
reasoning. 

\item Is it true that $(A \cap B) \cup C = A \cap( B \cup C)$?  Draw a
Venn diagram to justify your answer. Unsupported answers will receive
no credit.

\item  There are 5 people in the Smith family.  Use the multiplication
principle to explain why there are 120 ways that all of the members of
the Smith family may stand in a line for a family photograph.



\item Answer the following.  You must write out your answer using
factorials or writing out which integers you will multiply to obtain
the answer.  You do not need to give the numerical answer.

\begin{enumerate}
\item How many different 4 letter words can you form using the letters
`ABCDEFG'?. Assume that repetitions are not allowed.


\item How many distinguishable permutations are there of the letters
in `BOZOZ'?

\item How many permutations are there of the letters in `COURANT'?
\end{enumerate}


\item A die is rolled and a coin is flipped.  You are given \$1 if the
coin lands heads up.  You are also given \$2 if the die shows an odd
number.  What is the probability of winning at least \$1 when playing
this game one time?
\vspace{3 in}
\item
A coin is flipped five times.  What is probability of obtaining at
least one head?





\item You are to draw two balls without replacement from an urn which
contains 5 red and 4 blue balls.  
\begin{enumerate}
\item Draw a tree which summarizes this game.


\item Compute the probability that the second ball is red.


\item Compute the conditional probability that the first ball is blue
given that the second ball is red.


\end{enumerate}


\item You play a game where you are to roll a die 5 times.  If you
obtain 5 sixes, you will receive \$3000 and if you obtain exactly 4
sixes you will receive \$300.  What is the expected value of this
game?  Would it be profitable to play this game if you have to pay
\$1.50 for each play?




\end{flushleft}
\end{document}
