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\title{ESP Workshop, Worksheet \#11}
\date{\today}
\begin{document}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#11}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item Let's warm up with some lines and planes and surfaces. Let $P$
  be the plane defined by the equation $2x -y +3z =1$. That is, $P$ is the set of points $(x,y,z)$ in 3-space that satisfy the above equation. In set notation, this is:
\begin{displaymath}
P = \{ (x,y,z) \in \mathbb{R}^3 : 2x -y +3z =1 \}
\end{displaymath}
\begin{enumerate}
\item Find an equation for the plane through the origin that does not
  intersect $P$.

\item Find three distinct points in $P$. Use these points to find a
  normal vector for $P$.

\item Find two other planes $Q$ and $R$ such that $P\perp Q$, $Q\perp
  R$, and $P\perp R$.

\item Let the line $l$ be defined by the parametric equations:
\begin{displaymath}
x = 3-t,\ y= 2+t,\ z=5t
\end{displaymath}
Where does $l$ intersect $P$? Can you find a line that does not
intersect $P$? What about a line that is contained in $P$?

\item Now consider the coordinate planes $x=0$, $y=0$, and $z=0$.
  (What have we been calling these planes?) How do they intersect the
  surface $S$ given by the equation $4x^2 +9y^2 +36z^2 = 36$? What did
  Professor Durbin call these intersections in class yesterday? Can
  you use them to guess what $S$ looks like in three dimensions?

\end{enumerate}

\item Here are eight equations and eight pictures. Match the equations
  with the surfaces they define.
\begin{displaymath}
\textrm{(a)}\ y^2 -z^2 =1+x^2
\quad \textrm{(b)}\ x^2 = 1-2z^2
\quad \textrm{(c)}\ 2x^2 -2y^2 +2z^2 =2
\quad \textrm{(d)}\ 9x^2 +4y^2 +z^2 = 1
\end{displaymath}
\begin{displaymath}
\textrm{(e)}\ z^2 = y-2x^2
\quad \textrm{(f)}\ x^2/36 +y^2/9 +z^2/4 = 1/36
\quad \textrm{(g)}\ y + z^2=x^2
\quad \textrm{(h)}\ 0 = x^2 +2z^2 - y^2
\end{displaymath}

\newpage

\begin{figure} \label{boat}
\begin{center}
\psfrag{d1}{$20^\circ$}
\psfrag{d2}{$30^\circ$}
\psfrag{f}{255 N}
\includegraphics{worksheet11}
\caption{This is a picture of a boat for question 3.}
\end{center}
\end{figure}

\item A boat is pulled onto shore using two ropes, as shown in
  Figure~\ref{boat}. If a force of 255 N is needed, find the magnitude
  of force in each rope.

\item Yesterday, I was telling a friend about Colt McCoy's amazing 3rd
  quarter against Oklahoma, but my friend had forgotten what a
  ``football'' was.  However, he's good at math, so all I had to say
  was ``$x^2 +4y^2 + 4z^2 = 1$'' and then he remembered what sport I
  was talking about. (Keep in mind: he's good with both equations and
  inequalities.) How could you tell this guy about...
\begin{enumerate}
\item ...soccer?
\item ...hockey?
\item ...rugby?
\item ...frisbee?
\item ...horseback riding? (Describe the saddle.)
\item ...snowboarding in a half-pipe?
\item ...skateboarding in an empty pool?
\item (Challenge.) ...golf? (Yes, dimples and everything.)
\end{enumerate}

\item Each edge of a cubical box has length 1 m.  The box contains
  nine spherical balls---eight orange and one white---all with the
  same radius $r$. The center of the white ball is at the center of
  the cube and it touches the other eight balls, each of which is near
  one of the eight corners of the box.  Each of the orange balls
  touches three sides of the box. Thus, the balls are tightly packed
  in the box. Find $r$.

\item A plane is capable of flying at a speed of 180 km/h in still
  air. The pilot takes off from an airfield and heads due north
  according to the plane's compass. After 30 minutes of flight time,
  the pilot notices that, due to the wind, the plane has actually
  traveled 80 km at an angle $5^\circ$ east of north.
\begin{enumerate}
\item What is the wind velocity?
\item In what direction should the pilot have headed to reach the
  intended destination?
\end{enumerate}

\end{enumerate}

\end{document}