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\begin{center}
{\Large TEST 5

Math 1090 - 4}
\end{center}
\vspace{.1in}
You have 50 minutes to complete the test.
Show all relevant work (or else no credit).  
Make your work legible, if I can't read it, I can't grade it.  
Each problem is worth 10 points.
Use the back of the pages if necessary.
\vspace{.1in}

Name \rule{5in}{.2pt}

\vspace{.2in}

\begin{enumerate}

\item  Set up the simplex matrix used to solve the 
linear programming problem below.

DO NOT SOLVE, JUST SET UP THE SIMPLEX MATRIX. (Task A)

Maximize $f=3x+5y+11z$ subject to 
\begin{eqnarray*}
2x+3y+4z & \leq & 60 \\
x +4y+z & \leq & 48 \\
5x+y+z & \leq & 50
\end{eqnarray*}

\item  A simplex matrix is given below.
Manipulate the matrix (as discussed in class and in the book)
until you can read off the solution.
Assume that all variable are nonnegative.

DO NOT READ OFF THE SOLUTION, JUST GET TO THE POINT WHERE YOU COULD.
(Task B)

\[
\left[
\begin{array}{rrrrr|r}
1 & 5 & 1 & 0 & 0 & 25 \\
2 & 5 & 0 & 1 & 0 & 30 \\
\hline
-2 & -6 & 0 & 0 & 1 & 0
\end{array}
\right]
\]

\item  I have taken a linear programming problem and turned it into a simplex matrix.
This is the matrix on the left.
I then manipulated this matrix to get the solution.
This is the matrix on the right.
(Task C)
\[
\left[
\begin{array}{rrrrrrr|r}
2 & 7 & 9 & 1 & 0 & 0 & 0 & 100 \\
6 & 5 & 1 & 0 & 1 & 0 & 0 & 145 \\
1 & 2 & 7 & 0 & 0 & 1 & 0 & 90 \\
\hline
-2 & -5 & -2 & 0 & 0 & 0 & 1 & 0 
\end{array}
\right]
\longrightarrow
\left[
\begin{array}{rrrrrrr|r}
0 & 1 &   13/8 &  3/16 & -1/16 & 0 & 0 & 155/16 \\
1 & 0 & -19/16 & -5/32 &  7/32 & 0 & 0 & 515/32 \\
0 & 0 &  79/16 & -7/32 & -3/32 & 1 & 0 & 1745/32 \\
\hline
0 & 0 &   15/4 &  5/8  &  1/8  & 0 & 1 & 645/8 
\end{array}
\right]
\]

Answer the following questions
\begin{enumerate}
\item  How many variables did we start with and what are they?
\item  How many inequalities did we start with and what are they?
\item  What is the function to maximize?
\item  How many slack variables and what are they?
\item  What is the solution to the linear programming problem?
(What is the maximum value for the function and what do all the variables need
to be to get this max, including the slack variables.)
\end{enumerate}

\item
You spend some time researching VW camper vans and 
studying their acceleration.
You test the following model:
The time $t$, in \emph{minutes}, that it takes a Volkswagen camper van
to accelerate to $x$ miles per hour can be modeled by the equation:
\[
t = .01(-597.62+.1976x^2 +21.017x)
\]
How fast can the camper van go after 11 minutes?
(Your answer may not reflect the acceleration in your car, 
but remember that this is a VW.)

\item  Given cost and revenue functions, answer the questions.
\begin{eqnarray*}
C & = & 360+10x+.2x^2 \\
R & = & 50x-.2x^2
\end{eqnarray*}
\begin{enumerate}
\item  Is there a maximum or minimum profit?
\item  What level of production gives the maximum or minimum profit?
\item  What is the maximum or minimum profit?
\end{enumerate}




\end{enumerate}

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