\begin{homework}\
\label{homework:14}


\medskip

\noindent
{\bf Part A.} Basic problems:

\medskip

\exercise %1
Let $f(x) = \begin{cases}1 + 4 x -x^2 & \text{for $x\leq3$}\\
                          \disp\frac{x+5}{2}          &\text{for $x>3$}
            \end{cases}$

(a) Find the maximum value of $f(x)$ for $x$ between $0$ and $4$.

(b) Find the minimum value of $f(x)$ for $x$ between $0$ and $4$.

(c) Graph $f(x)$ to check your answers.

\exercise %2.
Find the dimensions of the rectangle of largest area having fixed perimeter
$P$.  (Compare with Example~\ref{eg:14.4} above.)

\smallskip

\exercise %3
A box with square base and no top is to hold a volume $V$.  Find (in terms
of $V$) the dimensions of the box that requires the least material for the
five sides.  Also find the ratio of height to side of the base.  (This
ratio will not involve $V$.)

\smallskip

\exercise %4.
You have $l$ feet of fence to make a rectangular play area alongside the
wall of your house.  The wall of the house bounds one side.  What is the
largest size possible (in ft${}^2$) for the play area?

\smallskip

\exercise %5.
Marketing tells you that if you set the price of an item at \$10
then you will be unable to sell it, but that you can sell 500 items for
each dollar below \$10 you set the price.  Suppose your fixed costs total
\$3000, and your marginal cost is \$2 per item.  What is the most profit
you can make?

\smallskip

\exercise %6.
Find the area of the largest rectangle that fits inside a semicircle of
radius $r$ (one side of the rectangle is along the diameter of the
semicircle).

\smallskip

\exercise %7.
For a cylinder with given surface area $S$, find the ratio of height to
base radius that maximizes the volume.

\smallskip

\exercise %8.
You want to make cylindrical containers of a given volume $V$ using the
least amount of construction material.  The lateral side is made from a
rectangular piece of material, and this can be done with no material
wasted.  However, the top and bottom are cut from squares of side $2r$, so
that $2(2r)^2=8r^2$ of material is needed (rather than $2\pi r^2$, which is
the total area of the top and bottom).  Find the optimal ratio of height to
radius.

\smallskip

\exercise %9.
Given a right circular cone, you put an upside-down cone inside it so that
its vertex is at the center of the base of the larger cone and its base is
parallel to the base of the larger cone.  If you choose the upside-down
cone to have the largest possible volume, what fraction of the volume of
the larger cone does it occupy?  (Let $H$ and $R$ be the height and base
radius of the larger cone, and let $h$ and $r$ be the height and base
radius of the smaller cone.  Hint: Use similar triangles to get an equation
relating $h$ and $r$.)

\smallskip

\exercise %10.
In Example~\ref{eg:14.9}, what happens if $w>v$ (i.e., your speed on sand
is greater than your speed on the road)?

\smallskip

\exercise %11.
A container holding a fixed volume is being made in the shape of a cylinder
with a hemispherical bottom.  (The hemispherical bottom has the same radius
as the cylinder.)  Find the ratio of height to radius of the cylinder which
minimizes the cost of the container if (a) the cost per unit area of the
bottom is twice as great as the cost per unit area of the side, and the
container is made with no top; (b) the same as in (a), except that the
container is made with a circular top, for which the cost per unit area is
1 1/2 times the cost per unit area of the side.

\smallskip

\exercise %12.
(a) A square piece of cardboard of side $a$ is used to make an open-top box
by cutting out a small square from each corner and bending up the sides.
How large a square should be cut from each corner in order that the box
have maximum volume? (b) What if the piece of cardboard used to make the
box is a rectangle of sides $a$ and $b$?  Note that the side $x$ of the
square to be cut from each corner must be less than $a/2$ and $b/2$, and in
this way you can eliminate one of the two solutions of $f'(x)=0$.  In
addition, use the second derivative test to be sure your value of $x$
really gives the maximum volume.

\smallskip

\exercise %13.
A window consists of a rectangular piece of clear glass with a semicircular
piece of colored glass on top.  Suppose that the colored glass transmits
only $k$ times as much light per unit area as the clear glass ($k$ is some
fraction).  If the distance from top to bottom (across both the rectangle
and the semicircle) is fixed, find (in terms of $k$) the ratio of vertical
side to horizontal side of the rectangle for which the window lets through
the most light.  For what range of values of $k$ does this question make
sense?

\smallskip

\begin{rightindent}{3.5in}
\exercise %14.
In the same situation as Problem 4, suppose that instead of a rectangular
shape you decide to make the play area in the form of a trapezoid, using
three sections of fence of length $l/3$ each (see the diagram at the
right).  Find the angle $\theta$ that the sides should be turned outward in
order for the enclosed area to be maximal, and find the maximum play area
of this form.  Is the maximum play area greater or less than in Problem 4?
\end{rightindent}
\hfill
%--- figure PG (max/min Problem 23)--------------------
%
\begin{psfigure}{1.95in}{1in}{figPG.ps}
\puttext{-45., 0}{\Small$\theta$}
\puttext{45., 0}{\Small$\theta$}
\end{psfigure}

\bigskip

\noindent
{\bf Part B.} Additional Practice (do these as soon as possible, certainly
before the final exam):

\medskip

\exercise %15.
A terrorist sees a plane flying away from him at 500 mph at a height of 2
miles.  At time $t=0$ it is flying over a point on the ground which is 1.5
miles from him.  At that instant he shoots a missile at the plane, but
because of nervousness and perhaps a subconscious sense of morality, he
mistakenly aims directly at the present location of the plane, rather than
slightly ahead.  For this reason he misses the plane.  The purpose of this
problem is to determine by how much.  Suppose the missile travels at 2000
mph, and {\bf neglect both air resistance and gravity}.  First find
parametric equations for both the plane and the missile.  Then find the
time at which the two objects are closest.  (Hint: Use the distance
formula, and save yourself unnecessary algebra by finding the time at which
the {\it square} of the distance is minimal.)  By how much does the missile
miss the plane?

\smallskip

\exercise %16.
You are designing a poster to contain a fixed amount $A$ of printing
(measured in square inches) and have margins of $a$ inches at the top and
bottom and $b$ inches at the sides.  Find the ratio of vertical dimension
to horizontal dimension of the printing on the poster if you want to
minimize the amount of posterboard needed.

\smallskip


\exercise %17.
The strength of a rectangular beam is proportional to the product of its
width times the square of its depth.  Find the dimensions of the strongest
beam that can be cut from a cylindrical log of radius $r$.

\smallskip

\exercise %18.
What fraction of the volume of a sphere is taken up by the largest cylinder
that can be fit inside the sphere?

\smallskip

\exercise %19.
The U.S.~post office will accept a box for shipment only if the sum of the
length and girth (distance around) is at most 108 in.  Find the dimensions
of the largest acceptable box with square front and back.

\smallskip

\exercise %20.
Find the dimensions of the lightest cylindrical can containing 0.25 liter
(=250 cm${}^3$) if the top and bottom are made of a material that is twice
as heavy (per unit area) as the material used for the side.

\smallskip

\exercise %21.
A conical paper cup is to hold a fixed volume of water.  Find the ratio of
height to base radius of the cone which minimizes the amount of paper
needed to make the cup.  Use the formula $\pi r\sqrt{r^2+h^2}$ for the area
of the side of a cone.

\smallskip
\exercise %22.
If you fit the cone with the largest possible surface area (lateral area
plus area of base) into a sphere, what percent of the volume of the
sphere is occupied by the cone?  Compare your answer with the answer
to Example 14.7.

\smallskip

\exercise %23.
Two electrical charges, one a positive charge A of magnitude $a$ and the
other a negative charge B of magnitude $b$, are located a distance $c$
apart.  A positively charged particle $P$ is situated on the line between A
and B.  Find where $P$ should be put so that the pull away from $A$ towards
$B$ is minimal.  Here assume that the force from each charge is
proportional to the strength of the source and inversely proportional to
the square of the distance from the source.

\smallskip

\exercise %24.
Find the fraction of the area of a triangle that is occupied by the largest
rectangle that can be drawn in the triangle (with one of its sides along a
side of the triangle).  Show that this fraction does not depend on the
dimensions of the given triangle.

\smallskip

\exercise %25.
How are your answers to Problem 5 affected if the cost per item for the $x$
items, instead of being simply \$2, decreases below \$2 in proportion to
$x$ (because of economy of scale and volume discounts) by 1 cent for each
25 items produced.

\smallskip

\exercise %26.
You are standing near the side of a large wading pool of uniform depth when
you see a child in trouble.  You can run at a speed $v_1$ on land and at a
slower speed $v_2$ in the water.  Your perpendicular distance from the side
of the pool is $a$, the child's perpendicular distance is $b$, and the
distance along the side of the pool between the closest point to you and
the closest point to the child is $c$ (see the drawing). 
\begin{rightindent}{2.95in}
\noindent
Without even stopping to do any calculus, you instinctively choose the
quickest route (shown in the diagram at the right) and save the child.  Our
purpose is to derive a relation between the angle $\theta_1$ your path
makes with the perpendicular to the side of the pool when you're on land,
and the angle $\theta_2$ your path makes with the perpendicular when you're
in the water.  To do this, let $x$ be the distance between the closest
point to you at the side of the pool and the point where you enter the
water.  Write the total time you run (on land and in the water) in 
\end{rightindent}
\hfill
%--- figure PH (max/min Problem 25)--------------------
%
\begin{psfigure}{2.5in}{2.0in}{figPH.ps}
\puttext{-20., 4.}{\Small$x$}
\puttext{40., 4.}{\Small$c-x$}
\puttext{-45., -20.}{\Small$a$}
\puttext{55., 20.}{\Small$b$}
\puttext{8., -28.}{\Small$\theta_{1}$}
\puttext{30,30}{\Small$\theta_{2}$}
\end{psfigure}
%------------------------------

\noindent
terms of $x$ (and also the constants $a,b,c,v_1,v_2$).  Then set the
derivative equal to zero.  The result, called ``Snell's law'' or the ``law
of refraction,'' also governs the bending of light when it goes into water.

\end{homework}

