\begin{homework}\
\label{homework:8}

\medskip

\exercise %1.
Convert to radians: (a) $15^\circ$, (b) $120^\circ$, (c) a quarter
revolution, (d) 10 revolutions.

\smallskip

\exercise %2.
A \index{ bicycle wheel} bicycle wheel has radius 40 cm.  Find the rate in rad/sec at which the
wheel is turning if the bicycle is traveling at (a) 10 m/sec, (b) 10 km/hr.

\smallskip

\exercise %3.
A \index{ drawbridge} drawbridge whose two spans are each 30 meters long is opened as shown
 below.  Express in terms of $\theta$: (a) the height $y$ of the
end of a span, (b) the slope of the left span, (c) the distance $x$ between
the ends of the spans.


\begin{center}
%---------------- figure IGa--------------------------
\begin{psfigure}{2in}{1.25in}{figIGa.ps}
\puttext{0,10}{$x$}
\puttext{-5,-25}{$y$}
\puttext{-70,-45}{$\theta$}
\end{psfigure}\\
%-----------------------------------------------------
\end{center}




\smallskip

\begin{rightindent}{2.95in}
\exercise %4.
Using the diagram at the right, show that
\[
    \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta \,.
\]
\smallskip
\exercise %5
Sketch the graph of each of the following functions:

(a) $y=1+\sin(\pi x)$, (b) $y=2\sin\left(\frac{\pi}{ 3}t\right)$,

(c) $y=2+2\sin\left(\frac{\pi}{ 2}(t-1)\right)$,

(d) $y=3+2\sin(2t+1)$.

\end{rightindent}
\hfill
%--------------------figur IGb-------------------------
\begin{psfigure}{2.5in}{1.5in}{figIGb.ps}
\puttext{-20,10}{\Small$1$}
\puttext{-20,-30}{\Small$\beta$}
\puttext{-20,-43}{\Small$\alpha$}
\puttext{35,20}{\Small$\alpha$}
\puttext{-45,-50}{\Small$A$}
\puttext{50,50}{\Small$F$}
\puttext{50,0}{\Small$E$}
\puttext{50,-50}{\Small$D$}
\puttext{13,-55}{\Small$C$}
\puttext{12,52}{\Small$B$}
\end{psfigure}



\bigskip
%-----------------------------------------------------


\smallskip

\exercise %6
The \index{ mean daily temperature} mean daily temperature in Fairbanks, Alaska was tabulated over a
30-year period, the average was taken over the 30 values for each day, and
the result was found to be very close to the sinusoidal function
$37\sin\left(\frac{2\pi}{ 365}(t-101)\right)+25$, where $t$ is the day of the
year.\footnote{This example is from an article by B.~M.~Lando and
C.~A.~Lando in {\it The \index{ Mathematics Teacher} Mathematics Teacher }, Sept.~1977.} (Here we are NOT
assuming 30 days per month.)  Sketch the graph of this temperature
function, find the maximum and minimum mean daily temperatures, and find
the warmest and coolest days of the year.

\smallskip

\exercise %7
Studies are made of the variation in population of \index{ foxes} foxes and \index{ rabbits} rabbits in a
certain large forest.  The number changes from year to year.  Roughly
speaking, when there are a lot of rabbits, the foxes will start to increase
in number, because they can eat rabbits plentifully; but then this
decreases the number of rabbits, causing the foxes to be short of food, and
this causes a decrease in the fox population; and, finally, when there are
fewer foxes the rabbits will be able to increase in number without being
eaten, so we are back to where we started.  Thus, we might expect a
sinusoidal oscillation in the number of foxes and in the number of rabbits.
(This is the so-called ``\index{ predator-prey} predator-prey'' problem in \index{ mathematical biology} mathematical biology.)
Suppose that $t$ stands for the number of years since the study began, $F$
stands for the number of foxes, and $R$ stands for the number of rabbits.
Suppose that $F$ and $R$ are each a sinusoidal function of $t$.  Use the
following table of data (which gives the maximum and minimum population of
each) to find a formula for $F$ and for $R$ in terms of $t$:

\begin{centering}
\begin{tabular}{ccccccc}
$t$ & &$0$&$1$&$2$&$3$&$4$ \\
$F$ & &$150$&$200$&$150$&$100$&$150$\\
$R$ & &$10000$&$7500$&$5000$&$7500$&$10000$
\end{tabular}\\
\end{centering}

\smallskip

\exercise %8
Write a formula for each of the following functions of $t$, assuming that
the function is sinusoidal (also suppose that a year consists of 12 months
of 30 days each):

(a) the mean low daily temperature, if the coldest is 20${}^\circ$F on
January 19 and the highest low temperature reading is 60${}^\circ$F on July
19;

(b) the time of sunset each day, if the sun sets at 9 p.m. on June 21 and
at 5 p.m. on December 21;

(c) the temperature in a well-insulated building, which varies between
25${}^\circ$C at 5 p.m. (which is 17:00 on the 24-hour clock) and
15${}^\circ$C at 5 a.m;

(d) the \index{ velocity} velocity of a weight attached to the end of a \index{ spring} spring, which is
extended and released at time $t=0$, if the weight reaches its maximum
velocity of 10 cm/sec after $0.1$ sec;

(e) the height above the ground of a \index{ pebble} pebble that is picked up in the tread
of a \index{ bicycle tire} bicycle tire at time $t=0$, if the tire has radius 40 cm and the
bicycle is traveling at 10 m/sec.

\end{homework}

