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\centerline{\textbf{\large Worksheet on optimization and related rates}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

This worksheet is designed to walk you through some optimization and
related-rate problems.  On the first two pages, I give you some general
guidelines for such problems.  On subsequent pages, I give you some problems
with substeps outlined for you.  (On an exam, of course, you won't have such
outlined substeps.  This is practice.)

(Disclaimer:  Some of these problems are adapted from Anton's \emph{Calculus},
3rd ed., Wiley, 1988.)

John Kerl \\
\texttt{kerl.john.r@gmail.com} \\
2008-11-20

\textbf{Guidelines for optimization problems}

\begin{enumerate}
\item Draw a picture and label quantities.
\item Write down any expression(s) you have which relate the quantities.
\item Write down an expression for the quantity you want to optimize.
	Remember that \emph{optimize} means (from its Latin root) \emph{make best}.
	The word problem may say \emph{minimize} or \emph{maximize}; the techniques
	are the same in either case.  (That's why we don't have separate textbook
	sections for minimization and maximization.)
\item Generally, that expression will depend on more than one variable.
	Use the relations from step (ii) to eliminate a variable.
\item Differentiate the quantity to be optimized with respect to the
	remaining variable.  Set that derivative to zero and solve.
\item Extrema occur at critical points (which you just found) or boundary
	points.  Find the value at boundary points.
\item The max/min occurs the smallest/largest of the critical point(s)
	and the boundary points.
\item Sanity-check your answer.
\end{enumerate}

\textbf{Example:}  Find the dimensions of the rectangle of maximum area with
perimeter 1000 feet.

\begin{enumerate}
\item Draw a picture of a rectangle, labeling the length $\ell$ and width $w$.
\item The perimeter is $2\ell+2w=1000$ and the area is $A=\ell w$.
\item The quantity to be optimized is $A=\ell w$.
\item This depends on $\ell$ and $w$.  Use the perimeter constraint
	to solve for $\ell$:  $\ell = 500-w$.  (You could also solve for $w$.)
	Then we have $A=\ell w = (500-w)w = 500w-w^2$.
\item The remaining variable is $w$.  Differentiating gives us
	$dA/dw = 500-2w$.  Setting this equal to zero we get
	$500-2w=0$ so $w=250$.
\item The smallest $w$ can be is zero; this gives a rectangle of zero
	area.  The largest $w$ can be (due to the perimeter constraint) is 500.
	In that case $\ell$ is zero, also giving zero area.
\item So, the maximum area occurs when $w=250$ (i.e. at the critical
	point, not at the boundary points).
\item Sanity check:  Using the perimeter constraint we get $\ell=250$
	as well.  The perimeter adds up to 1000; the length and width are all in a
	reasonable range (nothing is negative).  (Can you think of anything else to
	check?)
\end{enumerate}

%% ================================================================
\newpage

\textbf{Guidelines for related-rate problems}

\begin{enumerate}
\item Draw a picture.
\item Label quantities.  Note which ones can vary, and which ones are held
fixed.
\item Write down an expression relating the quantities.  You do \emph{not}
	have to solve for one variable in terms of the other.  (You'll solve
	for the derivative below.  This is kind of like implicit differentiation,
	where you solve for $dy/dx$ even if you can't solve for $y$.)  An equation
	relating the two quantities will suffice.
\item If there are more than 2 quantities which vary, use the additional
	information provided to eliminate variables until only two are left.
	(This step doesn't occur in every related-rate problem.)
\item Take the derivative of both sides (almost always with respect to time).
	Make sure to use the chain rule whenever you encounter a quantity which
	varies with time.  Remember that ``rate of change'' and ``how fast''
	in the word problem both mean ``take the derivative''.
	You will get an equation with two derivatives in it.
\item You will have been given the value of one derivative.  Solve for the
	unknown derivative.
\item Sanity-check your answer.
\end{enumerate}

\textbf{Example:} An initially empty spherical balloon is inflated at a rate of
2 cubic inches per second.  Find the rate of change of the radius of the
balloon when the radius reaches 5 inches.

\begin{enumerate}
\item Draw a picture of a sphere of radius $r$.
\item The volume $V$ and the radius $r$ both vary with time as the balloon
	inflates.
\item The volume of a sphere of radius $r$ is $V=\frac{4}{3}\pi r^3$.
	There is no need to solve for $r$.  We can solve for $dr/dt$ later.
\item There are only two varying quantities ($V$ and $r$), so there is nothing
	to eliminate.
\item Differentiating both sides, we have
	$dV/dt = 4 \pi r^2 dr/dt$.
\item We are given $r=5$ and $dr/dt|_{t=5}=2$.  Plugging these in and
	solving for $dr/dt$ we have
	$$dr/dt=\frac{dV/dt}{4\pi r^2} = \frac{2}{100 \pi} \approx 0.0064
	\;\textrm{in}^3/\textrm{sec}.$$
\item At radius $r=5$, the volume is
$4 \cdot 125 \pi/3 \approx 524 \textrm{in}^3$.
Only two more cubic inches of air will be added in the next second.  It makes
sense that the radius would be changing by very little.  Also, the radius is
increasing, not decreasing, so apparently we didn't make a minus-sign error.
\end{enumerate}

%% ================================================================
\newpage

\textbf{Problem 1.}
You are to make a box with square base and no top.  You will (as guided below)
find the dimensions that minimize the surface area of the box if the volume of
the box is to be 1000 cm${}^3$.

What do you call the three dimensions (front-to-back distance, left-to-right
distance, bottom-to-top distance) of the box?  The variable names are your
choice.

\vspace*{0.3in}

What is the volume of a box in terms of those three lengths?  Write an equation.

\vspace*{0.3in}

What does ``square base'' mean in terms of those three lengths?

\vspace*{0.3in}

Write an equation for $V=1000$, in terms of those lengths.  How many variables
does $V$ depend on?

\vspace*{0.3in}

What is the surface area $A$ of the box?  Remember we're not including the top.
There are four sides, and the bottom.  Those are all rectangles.  What is the
area of each?  Remember that areas add.  What is the sum of the 5 areas?

\vspace*{0.5in}

What is the name of the quantity we are asked to optimize?

\vspace*{0.3in}

How many variables does that quantity depend on?

\vspace*{0.3in}

Can those variables take on any value, or are they related somehow?
Write down an equation involving them.  Then solve that equation
for one of the variables.

\vspace*{0.5in}

Plug that variable back into the area equation.  Area should now depend on just
one variable.

\vspace*{0.3in}

Differentiate that equation and set it to zero.

\vspace*{0.5in}

Solve for the remaining variable.

\vspace*{0.5in}

Now find the other dimensions of the box.

\vspace*{0.3in}

Does everything look reasonable?  Why or why not?

%% ================================================================
\newpage

\textbf{Problem 2.} You are to fence a rectangular area.  The fencing for the
left and right sides costs \$2/foot; the fencing for the front and back sides
costs \$3/foot.  Find the dimensions of the rectangular area that result in
least cost, if the area is to be 3200 square feet.

I won't break down the steps for you on this one --- do it on your own.  Show
each step.

(Hint to get you started:  The cost of the fence is a concept somewhat related
to the perimeter of the fence.  If, for example, the depth is 10 feet and the
width is 320 feet, then it would cost $2\cdot 2\cdot 10 + 3 \cdot 2\cdot 320 =
1960$ dollars to make that fence.)

% * Draw a picture and label quantities.
% * Write down any expression(s) you have which relate the quantities.
% * Write down an expression for the quantity you want to optimize.
% * Generally, that expression will depend on more than one variable.
%	Use the relations from step (ii) to eliminate a variable.
% * Differentiate the quantity to be optimized with respect to the
%	remaining variable.  Set that derivative to zero and solve.
% * Extrema occur at critical points (which you just found) or boundary
%	points.  Find the value at boundary points.
% * The max/min occurs the smallest/largest of the critical point(s)
%	and the boundary points.
% * Sanity-check your answer.

%% ================================================================
\newpage

\textbf{Problem 3.}  Grain pours from a chute at a rate of 8 cubic feet per
minute.  It makes a conical pile with the property that the height of the pile
is always twice the radius.  (See the Wikipedia article on \emph{angle of
repose} if you'd like to find out why this is a reasonable assumption.)
How fast is the height changing when the height reaches 6 feet?

First I will tell you that the volume of a right circular cone of radius $r$
and height $h$ is $V=\frac{1}{3} \pi r^2 h$.  Which quantities in this equation
are varying with time?  Which are not?

\vspace*{0.7in}

I just wrote the equation for you in this case, so you don't need to.  Note
that we don't need to solve for $h$.  Are there more than two time-varying
quantities?  If so, how can you eliminate one of them?  Remember that the
question is asking about rate of change of height, so the height variable needs
to survive this elimination process.

\vspace*{0.7in}

Differentiate both sides of the equation with respect to time.

\vspace*{0.7in}

How many derivatives do you see?  Which one's value did I give you?  Which
one's value did I ask you to solve for?

\vspace*{0.7in}

Solve for the requested derivative.

\vspace*{0.7in}

Does your answer make sense?  Why?

% * Draw a picture.
% * Label quantities.  Note which ones can vary, and which ones are held fixed.
% * Write down an expression relating the quantities.  You do \emph{not}
%	have to solve for one variable in terms of the other.  (You'll solve
%	for the derivative below.)  An equation relating the two quantities
%	will suffice.
% * If there are more than 2 quantities which vary, use the additional
%	information provided to eliminate variables until only two are left.
%	(This step doesn't occur in every related-rate problem.)
% * Take the derivative of both sides (almost always with respect to time).
%	Make sure to use the chain rule whenever you encounter a quantity which
%	varies with time.  Remember that ``rate of change'' and ``how fast''
%	in the word problem both mean ``take the derivative''.
% * You will get an equation with two derivatives in it.  You will have
%	been given the value of one derivative.  Solve for the unknown derivative.
% * Sanity-check your answer.

%% ================================================================
\newpage

\textbf{Problem 4.}
An aircraft climbs at an angle of $30^\circ$ to the horizontal.  How fast is
the altitude changing if the air speed is 500 miles per hour?

I won't break down the steps for you on this one --- do it on your own.  Show
each step.

(Hint:  Draw a triangle with its base on the ground.  The altitude of the plane
is straight up.  Which direction is the plane flying in?  It rhymes with
\emph{flypotenuse}.)

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