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\centerline{MATH 135 -- Calculus 1}
\centerline{Review on Optimization}
\centerline{December 11, 2019}
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\noindent
{\it Practice Problems}
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\noindent
Solve the following optimization problems:
\begin{enumerate}
\item Let $x,y$ be two non-negative numbers such that $2x + y = 100$.  What is the maximum value of 
$xy^2$?
\item What is the maximum area of a rectangle inscribed in a $3-4-5$ right triangle?  Assume
the sides of the rectangle are parallel to the sides of length 3 and length 4.
\item All units in a 100-unit apartment building are rented out when the monthly
rent is set at $r = 900$ dollars per month.  Suppose that one unit becomes
vacant with each $10$-dollar increase in the monthly rent and each occupied unit
costs $80$ dollars per month in maintenance.  What rent $r$ will maximize the
monthly profit of the building landlord?
\item Your job is to design a rectangular industrial warehouse consisting 
of three separate rectangular rooms of equal areas formed by two interior walls
parallel to two of the exterior walls.  The wall materials cost
$500$ dollars per linear meter and the company allocates $2,400,000$ dollars
for the portion of the project consisting of the walls.  What dimensions 
will maximize the floor space (i.e. area) of the warehouse?
\item A billboard of height $b$ is mounted on an outside wall of building with 
its bottom at a distance $h$ from the pavement as in Figure 30 in our book.
At what distance $x$ from the wall should an observer stand
to make the billboard appear largest?  (That is, we want to maximize the angle
subtended by the billboard at the eye of the observer.)
\end{enumerate}
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