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\def\header{{\hfill Final Exam
\hfill A. Miller
\hfill Spring 2006
\hfill Math 131\hfill}}

\markboth\header\header

\newcount\probno
\probno=0
\def\prob{\advance\probno by 1 \par\bigskip\noindent\the\probno . }
\def\tprob {\advance\probno by 1 \par\bigskip\noindent\the\probno .
(True or False)$\;\;\;$ }

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\begin{document}

Do two of the problems (1-5) below. They are 10 points each.
Show all your work.  Explain your solution.  Put your answers on the answer
sheet.

\bigskip

\prob Find the area and perimeter of the polygon below.
Each square is 1 cm by 1 cm.

\begin{center}
\unitlength=1.00mm
\begin{picture}(100, 80)
\multiput(0,0)(0,10){9}{\line(1,0){100}}
\multiput(0,0)(10,0){11}{\line(0,1){80}}
\put(70,10){\line(2,3){20}}
\put(90,40){\line(-1,1){30}}
\put(60,70){\line(-2,-1){20}}
\put(40,60){\line(1,-2){10}}
\put(50,40){\line(-3,-1){30}}
\put(20,30){\line(5,-2){50}}
\end{picture}
\end{center}

\prob Prove Pythagoras's formula.

\prob A row boat is tied to a pier using a 10 foot rope. The rope is tied to a
cleat on the boat which is 1 foot above the water line. The other end is
attached to a pylon on the pier at a point 7 feet above the water. Find the
largest horizontal distance that the boat can be from the pier.

\prob A map of Lake Mendota is traced onto a 1 foot by 1 foot plywood square.
The square weighs 30 ounces.  The picture of the lake is then cut out and
weighed.  Assume that one acre of Lake's surface corresponds to 10 square
inches on the map.  If the the cut out weighs 20 ounces, how many acres is
Lake Mendota?

\prob A disk with diameter 8 inches is cut into half along its diameter.  Each
half disk is used to make the slanted side of a cone.  What is the volume
of these two cones?

\newpage

Put all your answers on the answer sheet. Each of these problems is one
point and there is no penalty for guessing.

\bigskip

\prob If a planar shape is formed by intersecting a vertical
or horizontal plane with a solid torus (i.e. a bagel or donut), then
it is
\begin{enum}
\item two disks
\item an annulus
\item a circle
\item it could be any of the above
\end{enum}

\prob If you round off $6252.084321$ to the nearest 100 then you get:
\begin{enum}
\item 6252
\item 6252.08
\item 6200
\item 6300
\end{enum}

\prob The area of triangle A is 20 and its height is 5.  If
triangle B is similar to triangle A but its height is 10, then
what is the area of B?
\begin{enum}
\item 20
\item 25
\item 40
\item 80
\end{enum}

\prob Two triangles $C$ and $D$ are similar.  The lengths of
the sides of triangle $C$ are 2,3, and 4.  The lengths of the sides
of triangle $D$ are 1,2, and ?
\begin{enum}
\item 3
\item 1.5
\item there is no such triangle $D$.
\item it is impossible to say because not enough information is given.
\end{enum}

\prob A soccer field is 120 feet by 90 feet.  What is the
area of this soccer field in square yards?
\begin{enum}
\item 10800
\item 1200
\item 3600
\item 210
\end{enum}

\prob  A hexagonal prism has
\begin{enum}
\item 8 faces, 18 edges, and 12 vertices.
\item 6 faces, 12 edges, and  8 vertices.
\item 8 faces, 12 edges, and  6 vertices.
\item 6 faces, 18 edges, and 14 vertices.
\end{enum}

\prob The {\bf surface area} of a box which is 2 feet high, 
3 feet wide, and 4 feet long is
\begin{enum}
\item 12 square feet
\item 24 square feet
\item 36 square feet
\item 52 square feet
\end{enum}

\prob Jupiter has many moons.  Three of them are
Europa, Ganymede, and Callisto.
$$\begin{array}{ccc}
\mbox{ moon }& \mbox{ distance from Jupiter in 1000 km }&
\mbox{ radius in km }\\
Europa & 670 & 1560\\
Ganymede & 1070 & 2631 \\
Callisto & 1882 & 2410 \\
\end{array}$$
For an observer on the surface of Jupiter which of the three moons
would appear to be largest?
\begin{enum}
\item Europa
\item Ganymede
\item Callisto
\item All three would be exactly the same size.
\end{enum}

\prob  Which of the following does {\bf not } imply that two
triangles are congruent:
\begin{enum}
\item ASA
\item SSS
\item SAS
\item AAA
\end{enum}

\prob 1 meter equals $\underline{\phantom{AAAAAAAAA}}$ inches
(answer rounded to nearest inch).
\begin{enum}
\item 100 \item 39 \item 36 \item 12
\end{enum}

\prob For a polygon on the integer lattice (its vertices have
integer coordinates), its area will be
${1\over 2}B+I-1$ where $B$ is the number of lattice points on
its perimeter and $I$ is the number of lattice points in its interior.
This is known as $\underline{\phantom{AAAAAAAAA}}$'s formula.
\begin{enum}
\item Archimedes
\item Cavalieri
\item Euler
\item Pick
\end{enum}

\prob A convex polyhedron satisfies the formula
 $$v+f=e+2$$
where $v$ is the number of vertices, $e$ is the number of edges,
and $f$ is the number of faces.
This is known as $\underline{\phantom{AAAAAAAAA}}$'s formula.
\begin{enum}
\item Archimedes
\item Cavalieri
\item Euler
\item Pick
\end{enum}

\prob The area of a triangle is $\sqrt{s(s-a)(s-b)(s-c)}$
where $a,b,c$
are the lengths of its sides and $s$ is half of its perimeter.  This
is called $\underline{\phantom{AAAAAAAAA}}$'s formula.
\begin{enum}
\item Archimedes
\item Euclid
\item Heron
\item Pythagoras
\end{enum}

\prob The ancient Greek mathematician, $\underline{\phantom{AAAAAAAAA}}$,
found that the volume of
a sphere is $4/3\pi r^3$ by looking at 
the hourglass (the enclosing cylinder minus the two inside cones).
\begin{enum}
\item Archimedes
\item Euclid
\item Heron
\item Pythagoras
\end{enum}

\prob Which of the following is {\bf not} an isometry of the plane?
\begin{enum}
\item rotation
\item reflection
\item shear
\item translation
\end{enum} 

\tprob Similar triangles are not congruent.
\tprob Doubling the perimeter of a square doubles its area.
\tprob If two rectangles have the same area, then they are congruent.
\tprob The surface area of a cube is always larger than its volume.
\tprob If two quadrilaterals have the same four angles, then they
are similar.

\newpage

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

\begin{center}
Circle the correct answer.
\end{center}

\newcount\xprobno

\def\spc{$\;$}
\def\xprob{\advance\xprobno by 1 \par\bigskip\noindent\the\xprobno .
\spc a \spc b \spc c \spc d \spc e}

\def\yprob{\advance\xprobno by 1 \par\bigskip\noindent\the\xprobno .
\spc True \spc False}

\begin{minipage}{1.5in}
\xprobno=5
\xprob\xprob\xprob\xprob\xprob
\end{minipage}
\begin{minipage}{1.5in}
\xprobno=10
\xprob\xprob\xprob\xprob\xprob
\end{minipage}
\begin{minipage}{1.5in}
\xprobno=15
\xprob\xprob\xprob\xprob\xprob
\end{minipage}
\begin{minipage}{1.5in}
\xprobno=20
\yprob\yprob\yprob\yprob\yprob
\end{minipage}

\bigskip\bigskip

Below and on the back of this sheet, put the
answers to 2 of the problems 1-5. Show all your work.  Explain your solution.
You may use additional paper if necessary.


\end{document}