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\noindent
{\bf Math 1030 Workshop \#5 (Due: July 27${}^{\mathrm{th}}$, 2015)} \hfill
{\bf Name: \underline{\hspace*{2.5in}}}

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\noindent
Please turn in a paper copy and {\bf SHOW YOUR WORK!}

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\begin{enumerate}
   \item Consider the function 
$\displaystyle{ f(x) = \left\{ \begin{array}{cc}
 \displaystyle -0.2x^2+x+3       & x \leq -1  \Large\strut \\
 \displaystyle (x-0.2)^2 e^{-0.3(x-0.2)^2} & -1 < x < 2 \Large\strut \\
 \displaystyle  2x^3-10.6x^2+18x-9  & x \geq 2  \Large\strut
 \end{array} \right. }$

Be careful! Wolfram Alpha may have trouble with this function. You might want to deal with it one piece at a time.

   \begin{enumerate}
   \item Find all of the critical points of $f(x)$.
             \qquad $x =$ \underline{\hspace{3in}}

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   \item Restricting our attention to the interval $[-2,3]$...
   
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         The maximum value of $f(x)$ is \underline{\hspace{1.25in}}. This occurs when $x=$ \underline{\hspace{1.25in}}.

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         The minimum value of $f(x)$ is \underline{\hspace{1.25in}}. This occurs when $x=$ \underline{\hspace{1.25in}}.
   \end{enumerate}

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  \item Green Villa Resort (located in Greenville, NC) uses golf carts to transport their guests around the grounds. Their carts cost \$2,000. They have noticed that a cart typically requires \$100 of repairs and maintenance during the its first year of operation, then \$360 during its second year, and then \$660 during its third year. 
  
Use Excel to find a power model for the {\bf average} annual repair costs. Then model the average annual cost of operating a golf cart using a function of the form: $\displaystyle{A(t) = \frac{C}{t} + Rt^r}$ where $C$ is the cost of purchasing the cart and $Rt^r$ models the repair costs. 
            
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\hspace{2in}            $A(t) =$ \underline{\hspace{4in}}
            
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             When $t=$ \underline{\hspace{1in}}, $A(t)$ is minimized. [Keep 5 decimal places.]
             
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            Green Villa should replace its golf carts every \underline{\hspace{1in}} years and \underline{\hspace{1in}} months. \\
            {}[Round up to the next whole month.]
            
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            If they do this, their average annual cost (per cart) should be \$ \underline{\hspace{1.5in}}.                       

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   \item Jonas manages an office which uses a lot of paper. He can get paper for \$3 a ream and has found that his average storage cost is \$0.05 per ream per year (base inventory costs on average inventory making all of the standard assumptions). Finally, Jonas pays \$30 every time he places an order. Let $C(x)$ be his annual cost function (for paper).
   
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   \begin{enumerate}
      \item If Jonas needs 1,000 reams each year, \quad $C(x)=$ \underline{\hspace{3in}}

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             His {\bf ideal} EOQ is \underline{\hspace{1in}} and his {\bf ideal} minimum annual cost is \underline{\hspace{2in}}.
   
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      \item If Jonas needs 10,000 reams each year, \quad $C(x)=$ \underline{\hspace{3in}}

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             His {\bf ideal} EOQ is \underline{\hspace{1in}} and his {\bf ideal} minimum annual cost is \underline{\hspace{2in}}.
   
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   \item Suppose that Jonas needs 5,000 reams each year. In addition, he found out that he gets a discount if he places a large order. For orders of 1,000 reams or more, he pays \$2.75 per ream. However, the paper company makes large deliveries in their ``big truck'', so the shipping cost jumps to \$200 and Jonas also figured that his inventory costs jump up to \$0.15 per ream (based on average inventory).

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             $\displaystyle{C(x) = \left\{ \Huge\strut \right.}$ 

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             His {\bf ideal} EOQ is \underline{\hspace{1in}}. His {\bf ideal} minimum annual cost is \underline{\hspace{2in}}.

   \end{enumerate}

\end{enumerate}

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