\def\wsn{14}
\input worksheet.tex

\noindent
{\bf INSTRUCTIONS}

\noindent
Do not answer the questions asked in the related rates problems given
below.  

\noindent
Instead, follow these instructions:

\medskip
\item{a)} Draw {\bf and label} a picture of the situation.

\medskip
\item{b)}  The rate(s) you know and the rate you are seeking should be
the time derivatives of quantities you have labeled.  State what those
quantities are.

\medskip
\item{c)}  Determine, an algebraic relationship involving the
quantities you have identified. 

\medskip
\item{d)}  Finally, venture a guess as to what type of answer you would
get.  Will it be positive or negative?  How would the rate depend on the
variables in the problem?

\bigskip\bigskip
\noindent
{\bf PROBLEMS}

\medskip
\item{1.}  Imagine the following magic triangle.  Its base is on a
horizontal surface and no matter what you do to its height, the
triangle always has area 10.  If you push down on the top of the
triangle so that it becomes shorter at a rate of 3 cm/sec, how fast
will the length of the base be changing when the triangle is 5cm tall?

\medskip
\item{2.}  Sophie and Elisa have made themselves two dimensional!
Sophie moves along the positive horizontal axis, and Elisa along the
graph of $f(x)=-\sqrt3x$, $x\leq0$.  At a certain time, Sophie is at
the point $(5,0)$ and moving with speed 3 units/sec; and Elisa is at a
distance of 3 units from the origin moving with speed 4 units/sec.  At
what rate is the distance between Sophie and Elisa changing?

\medskip
\item{3.}  For the following problem, assume that Ben is perfectly
spherical.  Suppose also that Ben melts at a rate proportional to his
surface area $a$ (i.e., ${dv\over dt}=ka$ for some negative constant
$k$.)  how fast is Ben's radius changing when his radius passes the 3
cm mark?  when his radius is 5 cm?  when his radius is $r$ cm?  (your
answers might involve the constant $k$.)

\medskip
\item{4.}  Rain is falling at the rate of $q$ inches per hour into
an open conical tank of height $h$ and radius $r$.  show that at each
instant the rate at which water is rising in the tank is
$$
q\times{(\hbox{area of tank openening})\over(\hbox{area of water
surface})}{\atop.} 
$$
next show that this holds for an open tank of arbitrary shape.

\medskip
\item{e.}  Amanda and Amanda are on a ferris wheel when the sun is
directly overhead.  the diameter of the wheel is 50 feet, and its
speed is 0.1 revolution per second.  (i) what is the speed of the
Amandas' shadow on the ground when they are at a two-o'clock position?
(ii) a one-o'clock position?  (iii) show that their shadow is moving
fastest when they are at the top or bottom, and its slowest when they
are at the three-o'clock and nine-o'clock positions.

\medskip
\item{f.}  A two-piece extension ladder leaning against a wall is
collapsing at the rate of 2 feet per second at the same time as its
foot is moving away from the wall at the rate of 3 feet per second.
How fast is the top of the ladder moving down the wall when it is 8
feet from the ground and the foot is 6 feet from the wall?

\medskip
\item{g.}  The speed limit on a stretch of highway is 55 mph.  Highway
patrol officer, Sgt. Miguel, stations himself at a point, out of view
of the motorists, 50 feet off the highway.  Miguel is equipped with a
radar gun which measures the speed at which a car approaches {\bf his
position}.  He takes a reading of suspected speeders by pointing his
radar gun at a point on the highway 120 feet from the point on the
highway closest to him.  The radar gun picks up a reading of 48
feet/sec for a green Chevy driven by Alyssa.  How fast is she
traveling?  Is Alyssa speeding?

\eject

\centerline{WAIT!  You don't get to leave yet!}

\bigskip
\noindent
{\bf MORE INSTRUCTIONS}

\medskip
\item{e)}  Differentiate the expression you found in part c) with
respect to time $t$.

\smallskip
\item{}  Plug in the appropriate values given for any variables
or rates then solve for the requested rate.






\bye