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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 5\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 5} 
\bigskip

\item{1.}  {\bf MacGyver Mathematics}

\itemitem{a)}  For the following task, you are equipped with a hacksaw
and a solar panel which is hooked up to a meter.  By laying flat
objects on the panel and reading the meter, you can determine how much
surface area the object spans on the panel.  Suppose you have a solid
object that you don't mind destroying. Describe a way to accurately
estimate its volume.

\smallskip
\itemitem{}  What could you do to make your estimate even more accruate?

\medskip
\itemitem{b)}  Say that you want to find the area of a certain
cross-sectional plane of a rock.  One way to find it is by sawing the
rock in two and measuring the area directly.  But suppose you do not
want to ruin the rock.  However, you do have a very accurate measuring
glass which gives you excellent volume measurements. How could you use
the glass to get a good estimate of the cross-sectional area?

\smallskip
\itemitem{}  What mathematical principle are you employing here?

\bigskip 
\item{2.}  Derive the volumes of the following solids using calculus:

\medskip
\itemitem{a)}  a ball of radius $r$;

\medskip
\itemitem{b)}  a right circular cone of radius $r$ and height $h$;

\medskip
\itemitem{c)}  a right circular cylinder of height $h$ and radius $r$.

\bigskip
\itemitem{3.\hskip12pt a)}  A deck of cards is stacked straight and
then pushed to the side so that the stack becomes skewed.  In which of
these two positions does the deck occupy more volume?

\medskip
\itemitem{b)}  Refer to question 2b.  What is the volume of a circular
cone of radius $r$ and height $h$ whose top point is not located
directly over the center of the base but is shifted $k$ units to one
side?

\medskip
\itemitem{c)}  Give both an intuitive and a mathematical argument to
support your answer to part b).

\bigskip
\item{4.}  Derive the volumes for the following solids:

\medskip
\itemitem{a)}  a right pyramid whose altitude is $h$ and whose base is
a square with sides of length $a$;

\medskip
\itemitem{b)}  the water which is two inches deep in a hemispherical
basin of radius one foot;

\medskip
\itemitem{c)}  a solid object with cross sections being squares of
side length $s(x)=\sqrt{\sin x}$;

\eject
\item{5.}  Suppose you have a tent which is supported by two flexible
aluminum poles which run along the ceiling of the tent from one corner
to the opposite corner of a square tent floor, crossing at the highest
point of the roof.  Assume that the shape that the poles make is a
parabola.
\medskip
\itemitem{a)}  If the tent has a square base 6 feet wide and is 6 feet
tall at the highest point, find an equation for the parabola formed by
the poles.  (Be sure to identify what the variables you are using
mean.)
\medskip
\itemitem{b)}  What shape are all of the horizontal cross-sections of
such a tent?
\medskip
\itemitem{c)}  What is the area of the cross-section 4 feet above the ground?
\medskip
\itemitem{d)}  What is the area of the cross-section $z$ feet above the ground?
\medskip
\itemitem{e)}  Set up an integral which equals the volume of the tent
and evaluate the integral.




\bye