\def\wsn{15}
\input worksheet.tex

\item{1.}  In each of the following equations, suppose that each
variable is actually a function of time $t$ and differentiate each
expression with respect to $t$.
$$\eqalign{
\pt a &x^2+y^2=100\hskip1in
\pt b {x+s\over5}={s\over1.5}\hskip.9in
\pt c 40y-xy=80\cr
\pt d &(x+7)(7-gt^2)=9x,\quad \hbox{where $g$ is a constant}\hskip.9in
\pt e V=kh^3,\quad \hbox{where $k$ is a constant}\cr}
$$

\medskip
\item{2.}  One sweltering $108^\circ$ day this past August, Josh and
Joel were cleaning the gutters of their elderly neighbor,
Mrs. Macinac, in repentence for having earlier hit a baseball through
her dining room window.  While Joel was perched atop a 10 foot ladder,
he made the mistake of angering Josh by accusing him of sleeping on
the job.  In retaliation, Josh began to pull the base of the ladder
away from the wall at a rate of ${1\over2}$ ft/sec.

\smallskip
\item{}  For the following questions, assume that Joel's balance is 
very good, and that the ladder was originally flat against the wall.

\medskip
\itemitem{a)} How far does Joel fall during his first four
seconds of motion?  The next four?  The next four?  The next four?
The last four?  (Use a calculator.)

\medskip
\itemitem{b)} From what you found in part a), what can you say about
the rate at which Joel is falling?

\medskip
\itemitem{c)} How fast is Joel approaching the ground when Josh
has pulled the bottom of the ladder 6 feet from the wall.

\medskip
\itemitem{d)} How fast is Joel moving when he hits the ground?
What is the physical significance of your answer?

\medskip
\itemitem{e)} The ladder, the wall, and the ground form a triangle.
How fast is the area of the triangle changing when Joel is 8
feet from the ground?  Is the triangle getting larger at this time, or
smaller? 

\bigskip
\itemitem{3.\hskip12pt a)}  A streetlight hangs 5 meters above the
ground.  Regina, who is 1.5 meters tall, walks away from the
point under the light at a rate of 2 meters per second.  How fast is
her shadow lengthening when she is 7 meters away from the point under
the light?
\itemitem{}  (Hint: Use similar triangles.)

\medskip
\itemitem{b)}  Suppose Regina has the ability to magically shrink
herself.  At what rate must she do this to keep her shadow a constant
length of 3 meters?  Write this as a function of only her distance
from the point under the light.

\medskip
\itemitem{c)} A light is on the ground 40 meters away from a building.
Chris, who is 2 meters tall, walks from the light toward the building
at 2 meters/second.  How rapidly is his shadow on the building growing
when he is 20 meters from the building?

\medskip
\itemitem{d)} A light is at the top of a pole which is 9 meters high.
A ball is dropped from height 7 at a point which is at a horizontal
distance 7 meters from the pole.  Assume that the ball falls according
to the law $s=gt^2$, where $t$ is the time in seconds, $s$ is the
distance in meters, and $g=4.9$ is a constant.  Find how fast the shadow
of the ball is moving along the ground at the time 4 seconds after it
is dropped.

\medskip
\item{4.}  Sand is flowing from a pipe at the constant rate of $s$
cubic meters per second, and is falling in a conical pile.  The
diameter of the base of this pile is always three times the altitude.
At what rate is the altitude of the pile increasing when the altitude
is $h$ meters?

\eject
\item{5.}  Many devices turn circular motion into linear motion or
vice versa by employing a version of the following device.  One end of
a rod is connected to a piston which is in a compartment which only
allows linear motion.  The other end of the rod is connected near the
outside of a wheel which is free to rotate around its center.  A
diagram of such a system is pictured below at four different times.
(This might be pistons and driveshaft in an automobile or the foot
pedal and wheel of a spinning wheel or some similar manually powered
device, for examples.)

\bigskip\hskip.5truein
\epsfxsize=5truein\epsfbox{/home/oehrtman/m210/pistons.eps}

\bigskip
\item{}  In the following suppose the rod in the diagram above has
lenght $l$ and is attached to the wheel at a point $R$ units from its
center.

\medskip
\itemitem{a)}  If the wheel spins at a constant rate and goes around
twice every second, how fast is the piston moving when it is half way
between its highest and lowest points?

\medskip
\itemitem{b)}  Suppose $l=8$ and $R=2$, and that at time $0$ the
piston is at its highest point.  Come up with a function which gives
the velocity of the piston  $t$ seconds later.  (What does the sign of
you velocity indicate?  How?)

\medskip
\itemitem{c)}  Now come up with a function which gives the position of
the piston $t$ seconds later.  (What did you pick for the height 0?)

\medskip
\itemitem{d)}  When is the piston moving fastest?



\bye