\begin{homework}\
\label{homework:12}

\medskip
{\bf Part A.} Basic problems:

\smallskip

\exercise %1.
A cylindrical tank has radius 20 cm.  How fast does the water level in the
tank drop when the water is being drained at 25 cm${}^3$/sec?

\smallskip

\exercise %2
A ladder 13 m long rests on horizontal ground and leans against a vertical
wall.  The foot of the ladder is pulled away from the wall at the rate of
0.6 m/sec.  How fast is the top sliding down the wall when the foot of the
ladder is 5 m from the wall?

\smallskip

\exercise %3.
A rotating beacon is located 2 miles out in the water.  Let $A$ be the
point on the shore that is closest to the beacon.  As the beacon rotates at
10 rev/min, the beam sweeps down the shore once each time it revolves.
Assume that the shore is straight.  How fast is the point where the beam
hits the shore moving at an instant when the beam is lighting up a point 2
miles downshore from the point $A$?

\smallskip

\exercise %4.
A baseball diamond is a square 90 ft on a side.  A player runs from first
base to second base at 15 ft/sec.  At what rate is the player's distance
from third base decreasing when she is half way from first to second base?

\smallskip

\exercise %5.
Sand is poured onto a surface at 15 cm${}^3$/sec, forming a conical pile
whose base radius is always half its altitude.  How fast is the altitude of
the pile increasing when the pile is 3 cm high?

\smallskip

\exercise %6.
A boat is pulled in to a dock by a rope with one end attached to the front
of the boat and the other end passing through a ring attached to the dock
at a point 5 ft higher than the front of the boat.  The rope is being
pulled through the ring at the rate of 0.6 ft/sec.  How fast is the boat
approaching the dock when 13 ft of rope are out?

\smallskip

\exercise %7.
A balloon is at a height of 50 meters, and is rising at the constant rate
of 5 m/sec.  At that instant a bicycle passes beneath it, traveling in a
straight line at the constant speed of 10 m/sec.  How fast is the distance
between them increasing 2 seconds later?

\smallskip

\exercise %8.
A cone-like vat has square cross-section (rather than circular
cross-section, as in the case of the usual cone).  The dimensions at the
top are 2 m $\times$ 2 m, and the depth is 5 m.  If water is flowing into
the vat at 3 m${}^3$/min, how fast is the water level rising when the depth
of water (at the deepest point) is 4 m?  Note: the volume of any conical
shape is (1/3) $\times$ height $\times$ area of base.

\smallskip

\begin{rightindent}{3.0truein}
\exercise %9.
The sun is setting at the rate of $1/4$ deg/min on a day when the sun
passed directly overhead.  How fast is the shadow of a 25 meter wall
lengthening at the moment when the shadow is 50 meters?  (See the picture.)

\smallskip
\end{rightindent}
\hfill
%--- figure ME --------------------------------------------------
\begin{psfigure}{2.45in}{1.25in}{figME.ps}
\end{psfigure}
%------------------------------

\smallskip

\exercise %10.
A woman 5 ft tall walks at the rate of 3.5 ft/sec away from a streetlight
that is 12 ft above the ground.  At what rate is the tip of her shadow
moving?  At what rate is her shadow lengthening?  Notice that in this
problem these rates do not depend on the instant in question --- both rates
are constant, provided that the woman's speed is constant.

\smallskip

\exercise %11.
A police helicopter is flying at 150 mph at a constant altitude of 0.5 mile
above a straight road.  The pilot uses radar to determine that an oncoming
car is at a distance of exactly 1 mile from the helicopter, and that this
distance is decreasing at 190 mph.  Find the speed of the car.

\smallskip

\begin{rightindent}{2.5in}
\exercise %12.
The trough shown on the right is constructed by fastening together three
slabs of wood of dimensions 10 ft $\times$ 1 ft, and then attaching the
construction to a wooden wall at each end.  The angle $\theta$ was
originally $30^\circ$, but because of poor construction the sides are
collapsing.  The trough is full of water.  At what rate (in ft${}^3$/sec) 
is 
\end{rightindent}
\hfill
%--- figure MF --------------------------------------------------
\begin{psfigure}{2.5in}{1.5in}{figMF.ps}
\puttext{20., -13.3}{\Small$10$}
\puttext{-53.3, -36.7}{\Small$1$}
\puttext{-80., -26.7}{\Small$1$}
\puttext{80., 23.3}{\Small$1$}
\puttext{-70, -20.}{\Small$\theta$}
\puttext{-41, -20.}{\Small$\theta$}
\end{psfigure}
%------------------------------
the water spilling out over the top of
the trough if the sides have each fallen to an angle of $45^\circ$, and are
collapsing at the rate of $1^\circ$ per second?

\smallskip

\begin{rightindent}{3.5in}
\exercise %13.
Suppose that in Example 12.6 the angle between the two roads is 120${}^\circ$,
instead of 90${}^\circ$ (that is, the South--North road actually goes in a
somewhat northwesterly direction from O).  In this problem use the law of
cosines: $c^2=a^2+b^2-2ab\cos\theta$.

(a) Express $\dot{c}$ in terms of $a,b,\dot{a},\dot{b}$.

(b) Notice that the angle $\theta$ is constant (and so $\cos\theta$ is also
constant).  State a story problem (perhaps involving a bug on a door,
rather than a car on a road) in which not only $a,b,c$ but also $\theta$
are all variables.  In that case suppose you know the values of
$a,b,\theta$ and their rates.  Find a formula for $\dot{c}$.
\end{rightindent}
\hfill
%- figure MK (cars on road intersecting at 120 degrees)--------------------
%
\begin{psfigure}{1.95in}{2in}{figMI.ps}
\puttext{-7.5, 25.}{\Small $A$}
\puttext{-9, 10.}{\Small $a$}
\puttext{17.5, 5.}{\Small $B$}
\puttext{10., -3.5}{\Small $b$}
\puttext{5., 15.}{\Small $c$}
\puttext{-7., -5.}{\Small $120^{\circ}$}
\end{psfigure}
%------------------------------

\bigskip

{\bf Part B.} Additional Practice (do these as soon as possible, certainly
before the midterm):

\smallskip

\exercise %14.
A light shines from the top of a pole 20 m high.  An object is dropped from
the same height from a point 10 m away.  How fast is the object's shadow
moving on the ground one second later?  (Neglect air resistance and take
$g=9.8$ m/sec${}^2$ for the falling object.)


\smallskip

\exercise %15.
The two blades of a pair of scissors are fastened at the point $A$.  Let
$a$ denote the distance from $A$ to the tip of the blade (the point $B$).
Let $\beta$ denote the angle at the tip of the blade that is formed by the
line $\overline{AB}$ and the bottom edge of the blade (see the diagram).
Suppose that a piece of paper is cut in such a way that the center
of the scissors at $A$ is fixed, and the paper is also fixed.  As the
blades are closed (i.e., the angle $\theta$ in the diagram is decreased),
the distance $y$ between $A$ and $D$ increases, cutting the paper.

\begin{rightindent}{3.0in}
(a)~Express $y$ in terms of $a$, $\theta$, and $\beta$.

(b)~Suppose you know the rate $d\theta/dt$ at which the angle between the
blades is decreasing.  Express $dy/dt$ in terms of $a$, $\theta$, $\beta$,
and $d\theta/dt$.

(c)~Suppose that the distance $a$ is 20 cm, and the angle $\beta$ is
$5^\circ$.  Further suppose that you are closing the scissors at the rate
of 50 deg/sec.  At the instant when $\theta=30^\circ$, find the rate (in
cm/sec) at which the paper is being cut.
\end{rightindent}
\hfill
%--- figure MG (scissors)---------------------------------------------
\begin{psfigure}{2.45in}{1.75in}{figMG.ps}
\puttext{-20., 5.}{$A$}
\puttext{-5., -2.5}{\Small$y$}
\puttext{65., 45.}{$B$}
\puttext{65., 0}{$C$}
\puttext{5., 5.}{$D$}
\puttext{-7.5, 3.5}{\Small$\theta$}
\puttext{30., 25.}{\Small$\beta$}
\end{psfigure}
%------------------------------


\smallskip

\exercise %16.
In the previous problem, 

(a) express in terms of $d\theta/dt$ the speed of the point on the scissors
that moves fastest (this point is $B$);

(b) if we have $a=20$ kilometers (i.e., we have scissors of grand
proportions) and if $\beta=0.001$ radians, find a value of $d\theta/dt$ for
which $dy/dt$ is a little faster than the speed of light ($c=300000$
km/sec) when the pair of scissors is almost closed;

(c) would the situation in your answer to part (b) be possible, or would it
violate the Theory of Relativity?  Explain.

\smallskip

\exercise %17.
In the situation of Example 12.6, find the rate at which the distance between
the cars is increasing or decreasing if

(a) car A is 300 meters north of O, car B is 400 meters east of O, both
cars are going at constant speed toward O, and the two cars will collide in
10 seconds;

(b) 8 seconds ago car A started from rest at O and has been picking up
speed at the steady rate of 5 m/sec${}^2$, and 6 seconds after car A
started car B passed O moving east at constant speed 60 m/sec;

(c) the functions $a(t)$ and $b(t)$ are given by the graphs below and you
want to find the rate at which the cars are separating at time $t= 2
\text{~sec}$.

\noindent {\bf Hint:} {Estimate derivatives by drawing tangent lines.}

%--- figure MH --------------------------------------------------
 
\begin{center}
\begin{psfigure}{\textwidth}{3in}{figMH.ps}
\end{psfigure}\\
\end{center}
%------------------------------

\begin{rightindent}{3.5in}
\exercise %18.
In Example 12.6 suppose that instead of car B you have a helicopter flying at
speed $v_2$ to the east of O.  Let $h$ be the altitude of the helicopter,
and let $c$ be the distance from car A to the helicopter.  Then the
three-dimensional Pythagorean Theorem states: $c^2=a^2+b^2+h^2$.  Find a
formula for $\dot{c}$ in terms of $a,b,h$ and the various rates if:

(a) the helicopter is flying at speed $\dot{b}$ at constant altitude;

(b) the helicopter is flying at horizontal speed $\dot{b}$ (i.e., the point
on the ground under the helicopter is moving at this speed), and at the
same time is gaining altitude at rate $\dot{h}$.
\end{rightindent}
\hfill
%--- figure MJ (car & helicopter)------------------------------------------
%
\begin{psfigure}{1.95in}{2.5in}{figMJ.ps}
\puttext{-5, 35.}{\Small $B$}
\puttext{-25., -26}{\Small $A$}
\puttext{-3, -5.}{\Small $h$}
\puttext{-20., 0}{\Small $c$}
\puttext{-24., -43.}{\Small $b$}
\puttext{-29., -31}{\Small $a$}
\end{psfigure}
%------------------------------

\end{homework}
