$(10$ points$)$ A small Coca$-$Cola factory servicing Georgia Tech must decide how much Coke Zero to produce each day. The cost of producing Coke Zero includes a $$200$ fixed cost associated with changing production lines, as well as a unit cost of $$\mathrm{.}5$ per liter of Coke Zero produced. Demand is modeled fairly accurately using a triangular distribution with the following pdf

$f(x)=\{\begin{array}{cc}\frac{2x}{c},& 0x50\\ \frac{200-2x}{c},& 50x100\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

Since Coca$-$Cola cares deeply about its loyal customer$-$base, the cont of unmet demand is estimated at $$1\mathrm{.}50$ per liter. The cost of storing any leftover Coke Zero is $$1$ per liter. Answer the following questions.

$($a$)(1$ point$)$ What is the value of $c?$

$($b$)(1$ point$)$ Derive the CDF of this distribution.

$($c$)(2$ points$)$ What is the optimal volume of Coke Zero to produce?

$($d$)(2$ points$)$ Suppose that we have $40$ liters of Coke Zero leftover from the day before. If we do not produce any additional Coke Zero, what is the expected cost?

$($e$)(2$ points$)$ Suppose that we have $40$ liters of Coke Zero leftover from the day before. If we produce an additional $30$ liters of Coke Zero, what is the expected cost?

$($f$)(2$ points$)$ Suppose that we have $40$ liters of Coke Zero leftover from the day before. What is optimal amount of Coke Zero to produce if we want to minimize the expected cost?

$5.(5$ points$)$ In one of our in$-$class activities, many students guessed that the optimal number of newspapers to order might be equal to the mean demand. After having solved many newsvendors problems, when would you expect that the optimal order quantity is "close" to the mean demand and when might it be very different from the mean demand?