\begin{homework}
\label{homework:15}\

\medskip

\exercise %1.
Find the fifth root of 7, using $x_0=1.5$ as a first guess.  Apply Newton's
method with three iterations.

\smallskip

\exercise %2.
A rectangular piece of cardboard of dimensions $8\times 17$ is used to make
an open-top box by cutting out a small square of side $x$ from each corner
and bending up the sides.  (See Problem 12 of the last homework set.) If
$x=2$, then the volume of the box is $2\cdot 4\cdot 13=104$.  Use Newton's
method to find the value of $x$ for which the box has volume 100.  Do your
computations using at least 3 significant figures of accuracy, and apply
Newton's method with enough iterations so that you know that your value of
$x$ is accurate to 3 significant figures.

%<------- changed 6x15 to 8x17 in problem above (Feb 24, 1993)-----%


\smallskip

\exercise %3.
A wheel of radius 1 picks up a pebble in its tread.  How much farther will
the wheel rotate (in radians) before the pebble has traveled a horizontal
distance of 1?  Use the cycloid formula, and take $\theta=2$ radians as a
rough guess.  Use Newton's method with three iterations.

\smallskip

\begin{rightindent}{3.25in}
\exercise %4.
Consider a vertical line passing through the point $(-\pi/2,-1)$ on the
curve $y=\sin x$.  This line is rotated clockwise until it rests against
the sine curve at a point somewhere between $(0.7,\sin 0.7)$ and $(0.8,\sin
0.8)$. (See figure at right.)  Use Newton's method to find the point where
the line rests, with enough iterations to be sure that your answer is
accurate to 4 significant figures.
\end{rightindent}
\hfill
%--- figure QB (Newton's method Problem 4)--------------------
%
%
\begin{psfigure}{2in}{2in}{figQB.ps}
\puttext{-20., -25.}{\Small$(-\frac{\pi}{2},-1)$}
\puttext{12.5, 17.5}{\Small ?}
\end{psfigure}
%------------------------------

\smallskip

\exercise %5.
In the baton example, notice that the tip of the baton makes a downward
loop at around $t=0.5$ sec.  Find the time (to four decimal places) when
the tip of the baton is at the bottom of the loop, and find its height at
that moment.

\smallskip

\exercise %6.
In the baton example, suppose that there is a wall located at exactly
$x=10$ ft.  Use Newton's method to find (to three decimal places) when the
tip of the baton hits the wall.


\smallskip

\begin{rightindent}{3.5in}
\exercise %7.
You are riding a double ferris wheel at an amusement park.  The large
circle has radius 40 ft, and the smaller circle has radius 10 ft.  (See the
diagram below.)  Your seat rotates clockwise around the smaller circle once
every 6 sec, and the center of the small circle rotates counterclockwise
around the big circle once every 20 sec.  At time $t=0$ you are at the
lowest possible point (50 ft below the center of the large circle).  Call
that point the origin.  First write parametric equations for your position
at time $t$.  Then use Newton's method to find (to at least three places
past the decimal point) the instant when you reach the highest point during
the first revolution of the large circle.  Also find your height at that
moment.
\end{rightindent}
\hfill
%--- figure QC (Newton's method Problem 7)--------------------
%
\begin{psfigure}{2.2in}{2.2in}{figQC.ps}
\put(12,-55){\vector(-2,1){10}}
\puttext{25, -60}{\Small Position at $t = 0$}
\end{psfigure}
%------------------------------

\smallskip
\noindent
\begin{rightindent}{3.5in}
\exercise %8.
Two mine shafts meet at an angle of 120${}^\circ$, as shown above.  The
horizontal shaft has a width of 7 ft, while the entrance shaft is 9 ft
wide.  What is the longest ladder that can negotiate the turn?  Neglect the
thickness of the ladder, and assume it is not tipped as it is maneuvered
around the corner.  At a given moment the ladder makes an angle of $\theta$
as shown.  First convince yourself that the problem can be rephrased as
follows: Find the angle $\theta$ at which the length $l=\overline{AB}$ is
minimal, and then find the corresponding value of $l$.  To do this, first
express $l=\overline{AC}+\overline{CB}$ in terms of
\end{rightindent}
\hfill
%--- figure QD (Newton's method Problem 8)--------------------
%
\begin{psfigure}{2.25in}{1.8in}{figQD.ps}
\puttext{-50., 5.}{\Small 7 ft}
\puttext{-15., -45.}{\Small 9 ft}
\puttext{35., -7.5}{\Small $120^{\circ}$}
\puttext{40., 5.}{\Small $\theta$}
\puttext{5., -12.5}{\Small $\alpha$}
\puttext{35., 20.}{\Small $B$}
\puttext{10., 5.}{\Small $C$}
\puttext{-15., -15.}{\Small $A$}
\end{psfigure}
%------------------------------
$\theta$ (notice that
$\alpha$ can be expressed in terms of $\theta$).  Then set $dl/d\theta=0$,
and use Newton's method with $\theta_0=30^\circ$ to find $\theta$ and $l$
to at least 3 significant figures of accuracy.
\end{homework}

