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\problemgrouptitle{Optimization Problems}

\begin{problemblock}{Solve the following optimization problems following the steps outlined in the text.\textbook{\\(See Appendix~\ref{ap:geometryformulas} for useful area and volume formulas.)}}

\item The entrance to a tent is in the shape of an isosceles triangle as shown below .  Zippers run vertically along the middle of the triangle and horizontally along the bottom of it.  If the designers of the tent want to have a total zipper length of 5 metres, find the dimensions of the tent that will maximize the area of the entrance.  Also find this maximum area.
\begin{center}
\resizebox{5.0cm}{!}{\includegraphics{graphics/tent}}
\end{center}
\answer{Base length=$\frac{5}{2}$~m, Height=$\frac{5}{2}$~m, Area=$\frac{25}{8}$~m$^2$}

\item Find the point on the line $y=2x+2$ closest to the point~$(3,2)$ by
\begin{enumerate}[label=(\alph*)]
\item\label{itm:closestptopt} Using optimization.
\item\label{itm:closestptint} Finding the intersection of the original line and a line perpendicular to it that goes through~$(3,2)$~.
\end{enumerate}
\answer{$\left(3/5,16/5\right)$~.  For part~\ref*{itm:closestptopt} note it is easier to minimize the distance-squared than the distance.  For part~\ref*{itm:closestptint} the line perpendicular is $y=-\frac{1}{2}(x-3)+2=-\frac{1}{2}x+\frac{1}{2}$~.  To find the intersection of this and the original line we solve the two equations simultaneously since the point of interest must lie on both lines.}

\item The product of two positive numbers is 50. Find the two numbers so that the sum of the first number and two times the second number is as small as possible.
\answer{First number=$10$, Second number=$5$}

\item A metal cylindrical can is to be constructed to hold 10 cm$^3$
  of liquid. What is the height and the radius of the can that
  minimize the amount of material
  needed? \answer{Radius$\displaystyle=\sqrt[3]{\frac{5}{\pi}}\approx1.17$~cm,
  Height$\displaystyle=2\sqrt[3]{\frac{5}{\pi}}\approx2.34$~cm}


\item A pair of campers wish to travel from their campsite along the river (location $A$) to visit friends 10~km downstream staying in a cabin that is 2~km from the river (location $B$) as shown in the following diagram:
\begin{center}
\resizebox{10.0cm}{!}{\includegraphics{graphics/canoe}}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item\label{canoeoptdownstream} If the pair can travel at 8~km/h in the river downstream by canoe and 1~km/h carrying their canoe by land, at what distance $x$ (see diagram) should they depart from the river to minimize the total time $t$ it takes for their trip?
\item\label{canoeoptupstream} On the way back from the cabin they can only travel at 4~km/h in their canoe because they are travelling upstream.  What distance $x$ will minimize their travel time in this direction?
\item\label{canoewconstants} Using the symbolic constants $a$ for the downstream distance, $b$ for the perpendicular land distance, $w$ for the water speed and $v$ for the land speed, find a general expression for optimal distance $x$.  Verify your results for parts~\ref*{canoeoptdownstream} and~\ref*{canoeoptupstream} of this problem by substituting the appropriate constant values.
\item Does your general result from part~\ref*{canoewconstants} depend at all on the downstream distance $a$? Discuss.
\end{enumerate}
\answer{\begin{enumerate}
\item $\displaystyle x=\frac{2}{3\,\sqrt{7}}\approx 0.252$~km
\item $\displaystyle x=\frac{2}{\sqrt{15}}\approx 0.516$~km
\item $\displaystyle x=\frac{b}{\sqrt{\left(\frac{w}{v}\right)^2-1}}$
\item Although the total time taken $t$ does depend on $a$ the value for $x$ does not.  As expected, it does not matter how far upstream you start, you would still turn off at the same location.  That said, $a$ does play a role in the solution because valid values of $x$ must lie in the interval $[0,a]$.  If $a$ had been smaller than the solution for $x$, say in part~\ref*{canoeoptdownstream} had the starting distance been $0.2$~km, then the critical number would no longer be in the interval and that solution would be invalid.  One would consider the endpoints of the interval to see which was optimal.
\end{enumerate}
}

\item A construction company desires to build an apartment building in the shape of a rectangular parallelpiped (shown) with fixed volume of $32000$~m$^3$. The building is to have a square base.  In order to minimise heat loss, the total above ground surface area (the area of the four sides and the roof) is to be minimised.  Find the optimal dimensions (base length and height) of the building.
\vspace{2em}
\begin{center}
\resizebox{3.0cm}{!}{\includegraphics{graphics/building}}
\end{center}
\answer{$x=40$ m, $h=20$ m}

\item A rectangular field is to be enclosed and then divided into three equal parts using 32~metres of fencing. What are the dimensions of the field that maximize the total area?
\answer{\raisebox{-2.5cm}{\resizebox{5cm}{!}{\includegraphics{graphics/field}}}}

\item Two power transmission lines travel in a parallel direction (north-south) 10~km apart.  Each produces electromagnetic interference (EMI) with the one to the west producing twice the EMI of that of the one to the east due to the greater current the former carries.  An amateur radio astronomer wishes to set up his telescope between the two power lines in such a way that the total electromagnetic interference at the location of the telescope is minimized.  If the intensity of the interference from each line falls off as $1/\mbox{distance}$, how far should the telescope be positioned from the stronger transmission line?
\answer{$\displaystyle 10\left(2-\sqrt{2}\right)\approx 5.9$ km }

\item  An athletic field consists of a rectangular region with
 a semicircular region at each end. The perimeter of the entire
 athletic field has to be 400 metres. Find the dimensions that
 maximize the area of the rectangular region.
\begin{center}
\resizebox{4.5cm}{!}{\raisebox{-4em}{\includegraphics{graphics/field2}}}
\end{center}
\answer{$x=100$ m, $\displaystyle y=\frac{200}{\pi}$ m}

\lab{\clearpage}
\item An antenna is to be created by bending a wire of length $16$~cm into the shape of a sector of a circle as shown.  In order to maximize the electric flux through the wire, the area of the sector is to be maximized. Find the dimensions $r$ and $\theta$ that will maximize the area.  Also find the maximum area.
\begin{center}
\resizebox{4.5cm}{!}{\raisebox{-4em}{\includegraphics{graphics/antenna}}}
\end{center}
\answer{$r=4$ cm, $\theta=2\ \mbox{rad}\approx115^\circ$, $A=16$~cm${}^2$}  

\item A \emph{tipi} is a conical tent, typically having no floor, constructed by placing skins, cloth, or canvas over a frame of poles.  In addition to its practical use as a shelter, it is an important symbol of the First Nations of the North American plains.
  \begin{enumerate}
  \item Assuming a large number of poles, a tipi is approximately a right circular cone with base radius $r$ and height $h$.  If a tipi has a fixed lateral surface area of $A$, find the radius and height in terms of $A$ that maximize its volume.  What is the ratio $h/r$ for the optimal tipi?
  \item An actual tipi having $n$ poles will have a conical base that is a regular $n$-sided polygon rather than a circle.  Repeat part~(a) by finding the apothem $H$ of the regular polygon base and height $h$ of the tipi that maximize its volume for fixed lateral surface area $A$.  Find the ratio $h/H$ for the optimal tipi. Show that in the limit $n\to\infty$ you recover the solution from part~(a). 
  \end{enumerate}
  \answer{\begin{enumerate}
    \item Radius $\displaystyle r=\sqrt{\frac{A}{\pi\sqrt{3}}}$, Height $\displaystyle h=\sqrt{\frac{2A}{\pi\sqrt{3}}}$, $\displaystyle \frac{h}{r}=\sqrt{2}$
    \item Let $\displaystyle\alpha=n\tan\frac{\pi}{n}$.  Apothem $\displaystyle H=\sqrt{\frac{A}{\alpha\sqrt{3}}}$, Height $\displaystyle h=\sqrt{\frac{2A}{\alpha\sqrt{3}}}$, $\displaystyle \frac{h}{H}=\sqrt{2}$

      As $n\to\infty$ the apothem $H$ approaches the radius $r$ of the circle.  The limit \[\lim_{n\to\infty}\alpha=\lim_{n\to\infty}n\tan\frac{\pi}{n}=\lim_{n\to\infty}\pi\frac{1}{\cos(\pi/n)}\frac{\sin(\pi/n)}{(\pi/n)}=\pi\cdot\frac{1}{1}\cdot 1=\pi\,,\]
        since $\pi/n\to0$ as $n\to\infty$, thus showing part (b) gives part (a) in the large $n$ limit.  Note that from observation a typical tipi has a height/radius ratio of about $\sqrt{2}\approx 1.4$\,.
    \end{enumerate}}
\end{problemblock}

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