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\centerline{MATH 135 -- Calculus 1}
\centerline{Applied Optimization Problems}
\centerline{November 22 and 25, 2019}
\vskip 10pt
\noindent
{\it Background}
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\noindent
In today's video, we discussed some first examples of optimization or
``max/min'' problems. Many of these come from potentially realistic
applications. A general strategy for solving these is:
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\begin{itemize}
\item Step 1: Draw pictures illustrating several different possible
    solutions of the problem, if appropriate.
\item Step 2: Identify which quantities are changing in your different
    solutions (the variables), and name them. Write down the quantity to
    be maximized or minimized, giving it a name too.
\item Step 3: (If there is more than one variable), write down any
    relations between the variables, and use the relation to solve for
    all variables in terms of one of them.
\item Step 4: Substitute from results of Step 3 into the function giving
    the quantity from Step 2. The goal here is to obtain a function of
    \emph{only one} variable.
\item Step 5: Find the critical points of the function from Step 4.
\item Step 6. Classify critical critical as local maxima or local minima
    using First or Second Derivative Tests. If the variable is limited
    to an interval, determine the function values at the endpoints, and
    find the overall maximum or minimum as indicated in the problem.
    (Note: ``largest, biggest, greatest, etc.'' in the statement of the
    problem usually means you are looking for a maximum value, while
    ``smallest, least, cheapest, etc.'' usually indicates you are
    looking for a minimum.)
\item Step 7. Find the maximum or minimum and write down the final answer.
    (And, of course, be sure you are answering the question that was
    asked!) 
\end{itemize}
Today and next time, we want to practice using this on several examples.
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\noindent
{\it Note:}  Unlike what has happened on the previous daily worksheets, I will ask 
you to hand in your work on this one by the end of the class on Monday, November 25
and I will grade it and return it on Monday, December 2.  The score will
be included in the Daily Quiz grade total, since we will not have a daily
quiz to be completed for Monday November 25.  Also, the third midterm
exam {\it will cover} this material.
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\noindent
{\it Discussion Questions}
\vskip 10pt
\begin{enumerate}
\item[A)]  Daphnis Aipolos, a Greek goatherd, wants to fence in a rectangular 
    pasture for his prize goats.  He has 1600 meters of fencing to use, and he can
    use the straight-line section of the sea shore bordering his master's land as 
    one of the  boundaries of the pasture (the goats don't like the water and won't 
    try to swim away in the sea).  What dimensions should the pasture be to maximize
    the area enclosed for the goats?
\item[B)]  A metal can with a top is to be manufactured to contain 30 cubic
    inches of ``Soprano Family Jersey Special'' pizza sauce. The can will 
    have the shape of a right circular cylinder. Find the dimensions (height and
    radius of the cylinder) that will \emph{minimize the cost of the can},
    assuming the metal for the top costs 1 cent per square inch, 
    but the metal for the bottom and the curved side wall of the can costs 
    2 cents per square inch because of a special acid-resistant coating
    used to increase the shelf-life of the pizza sauce.   (The inside top of the can
    does not need the acid-resistant coating because the cans are never completely filled with
    the sauce!)
\item[C)]  For some species of birds, it takes more energy to fly over water
    than over land (maybe because over land they can make use of updrafts). A lesser
    tufted grebe (this is a species of bird) leaves an island 5 km from
    point $A$, the nearest point to the island on a long straight shore.
    The grebe's nest is at point $B$, 13 km along the shore from point
    $A$. If it takes 1.4 times as much energy to fly one km over water
    as it does to fly one km over land, where on the shoreline should
    the bird head first in order to \emph{minimize the total energy} needed for
    the flight from the island to the nest.
\item[D)] Northern Iowa State Agricultural and Veterinary Junior College is
    building a new running track for their prize-winning track team --
    the ``Flying Farmers.'' The track is to be the perimeter of a region
    obtained by putting two semicircles on opposite ends of a rectangle,
    and that perimeter should be $440$ yards in length. Due to budget
    problems, the administration has decided to grow sweet corn in the
    area enclosed by the track and sell it to a local grocery store to
    generate some extra revenue. Determine the dimensions to build the
    track in order to \emph{maximize the area for growing corn}.  ({\it Careful}:  This is 
    a slightly ``tricky'' one in that the maximum is actually at an endpoint of the interval
    of possible values of one of the ``obvious'' variables.)
\end{enumerate}
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