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\noindent {\bf Problem statement} Example 1 in section 3.11 of the
text analyzes the following problem:

\smallskip

{\parindent=.3in \narrower\noindent
A 16-ft ladder leans against a wall. The bottom of the ladder is
5 ft from the wall at time $t=0$ and slides away from the wall at a
rate of 3 ft/s. Find the velocity of the top of the ladder at time
$t=1$. 

}


\smallskip


\noindent a) The textbook response to this question is
$-\sqrt{3}\approx -1.732$ ft/s. The minus sign means the top of the
ladder is sliding {\it down}. Check that the textbook's answer is
correct.

\smallskip

\noindent b) The speed of sound at sea level using the standard
atmosphere is about 340.29 meters per second. There are 3.280840 feet
in one meter. Using the assumptions of this model, find the angle
between the ladder and the ground at the time that the top of the
ladder breaks the speed of sound.

\smallskip

\noindent c) The speed of light is about 299,792,458 meters
per second. There are still 3.280840 feet in one meter. Using the
assumptions of this model, find the angle between the ladder and the
ground at the time that the top of the ladder moves at the speed of
light.

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\noindent {\bf Useful story for users of mathematics} Several people
are in a hot-air balloon, trying to land over a fog-shrouded
countryside at the end of a long day. The balloon dips low and they
see the ground faintly. One of them calls down to the ground, ``Where
are we?'' Some minutes later the wind is carrying them away and they
hear faintly, ``You're in a balloon!'' One person in the balloon
gondola says thoughtfully to the other, ``It's so nice to get help
from a mathematician.'' The other says, ``How do you know that was a
mathematician?'' The first replies, ``There are three reasons: it took
a long time get the answer, it was totally correct, and, finally, it
was absolutely useless.''