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\noindent
{\bf Math 1030 Workshop \#6} \hfill
{\bf Name: \underline{\hspace*{2.5in}}}

\vspace{0.05in}

\noindent
{\Large \bf DUE: Thursday, July 30${}^{\mbox{st}}$} Please turn in a paper copy and
{\bf SHOW YOUR WORK!}

\vspace{0.05in}

\begin{enumerate}
   \item  Apple has rolled out its latest product\dots{}the iPen. The local iStore has collected the following iData about their iProduct's demand:

\hfill
\begin{tabular}{|c||c|c|c|c|} \hline
iPens Sold \varspace{0.25in}{0.01in} & 15   & 100  & 200 &  350 \\ \hline
            Price \varspace{0.25in}{0.01in} & \$1,000 & \$225 & \$100 & \$50 \\ \hline
\end{tabular}  
\hfill \hfill

\begin{enumerate}[(a)]
   \item Compute elasticity (round to 3 decimals places):

\begin{tabular}{|c||c|c|c|c|} \hline
\hspace*{0.05in} iPens Sold \varspace{0.25in}{0.01in} & 15   & 100  & 200 &  350 \\ \hline
            Price \varspace{0.25in}{0.01in} & \qquad \$1,000 \qquad{} & \qquad{} \$225 \qquad{} & \qquad \$100 \qquad{} & \qquad \$50 \qquad{} \\ \hline
\end{tabular}  

\vspace{-0.05in}
    
\begin{tabular}{|c|c|c|c|c|} \hline
 \mbox{\parbox[c][0.4in]{1.455in}{Elasticity}} & \varspace{0.2in}{0.9in} & \varspace{0.2in}{0.9in} & \varspace{0.25in}{0.825in} & \mbox{\parbox[c][0.2in]{0.32in}{ $\times\mbox{ }\times$\\$\times\mbox{ }\times$ }} \\ \hline
\end{tabular}

\vspace{0.1in}

   \item Model this demand price (in Excel) using an {\bf power} trendline. 

\vspace{0.15in}

            $p_d(q) =$ \underline{\hspace{2in}}

\vspace{0.15in}

           According to this model, the store will sell \underline{\hspace{1.5in}} iPens if they set their price at \$75. 

\vspace{0.15in}

           When $p_d(q) = \$75$, our point elasticity (using this model) is $\varepsilon =$ \underline{\hspace{1.25in}} (round to 3 decimal places).

\vspace{0.15in}

          \underline{Circle the correct answer:} \qquad The store is currently charging \$75 for an iPen and wants to {\bf increase} its revenue, they should \varspace{0.5in}{0.01in}
         \quad {\Large raise \quad / \quad lower} \quad their price.
        
\vspace{0.05in}

% Bad model -- no maximum. :(
         \sout{What quantity and price will {\bf maximize} revenue? \quad $q =$ \underline{\hspace{1in}}  \quad $p=$ \underline{\hspace{1in}}}\\
         {\color{red}{}[There is \underline{\bf no maximum} for the corresponding revenue function!]}
        
\end{enumerate}

\vspace{0.1in}

   \item Suppose that some items point elasticity is $\varepsilon = 1.5$ at some price. If the price is lowered by 5\%, \dots
   
   \vspace{0.25in}
   
   the quantity is {\Large \quad lowered \quad / \quad raised \quad} by \underline{\hspace*{2in}} 
      
   \vspace{0.25in}

   and revenue is {\Large \quad lowered \quad / \quad raised \quad} by \underline{\hspace*{2in}}. 
   
   [Both answers should be percentages rounded to 3 decimal places.]

\vspace{0.5in}

\vspace{0.1in}
              
   \item Kyle is painting a very odd wall. Its height varies quite a lot. He needs to estimate the square footage of the wall so he can figure out how much paint to buy. Suppose that the wall is 30 feet long. Let $x$ be the number of feet from the start of the wall and $y$ be the height (in feet) of the wall as measured by Kyle\dots
   
   \begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \hline
   $x=$ & 0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 27 & 30 \\ \hline
   $y=$ & 5 & 7 & 8 & 7 & 10 & 9 & 6 & 5 & 4 & 6 & 7 \\ \hline
   \end{tabular}

\vspace{0.1in}

Approximate the area of this wall in 2 different ways: (1) Using a right hand rule approximation with $n=10$ rectangles and (2) Using Simpson's rule with $n=10$. [Round each answer to 4 decimal places.]

\vspace{0.15in}

Right hand rule: \underline{\hspace{1in}}  \hfill
Simpson's rule: \underline{\hspace{1in}}  \hfill

\vspace{0.15in}

\newpage

   \item Suppose that $R(t)=(t+0.5)^3e^{-t}$ models the construction rate of micro-houses (in millions of houses built per year). $t$ is the number of years since January 1, 2015. 
   
   \vspace{0.25in}
   
   When (after the initial peak) will the construction rate drop to 10,000 houses per year? $t=$ \underline{\hspace{1.75in}}\\ 
   \force \hspace{5.35in} [Round to 3 decimal places.]
   
   \vspace{0.25in}
   
   How many micro-houses will be built after January 1, 2025? \underline{\hspace{2in}}.\\
      
   \vspace{0.25in}
   
   When will the 5,000,000${}^{\mathrm{th}}$ micro-house be completed? \underline{\hspace{2in}}. \\
   \force \hspace{3.5in} [Give the month and year.]
      
   \vspace{0.1in}
   
   {\sc Alpha} Commands used / integrals computed:
   
   \vfill
   
   \item After surveying the residents of Raccoon City, we have found that the lifespan of an average citizen is 35 years with a standard deviation of 3. Assume that these lifespans are normally distributed.
   
   \vspace{0.25in}

    What percentage of residents live between 20 and 40 years? \underline{\hspace{2.5in}}\\
   \force \hspace{3.65in} [Answer in the form: XXX.XXX\%]
  
   \vspace{0.25in}
  
   If Raccoon City has 10,000,000 residents, how many live more than 50 years? \underline{\hspace{1.75in}}
  
   \vspace{0.25in}
  
   What is the cut-off determining the 10\% with the shortest lifespans? \underline{\hspace{2in}}\\
    \force \hspace{4.25in} [Round to 3 decimal places.]
   
    
   \vspace{0.1in}
   
   {\sc Alpha} Commands used / integrals computed:
   
   \vfill
  
\end{enumerate}

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