QUESTION $8$

$*$This is a new problemen

Q$3)$ You are in charge of the annual banquet for the Football Boosters Club at your college. One week before the banquet, you must make a commitment for the number of dinners. The price for these dinners will be $$7$ each. If fewer people show up than the committed number, you are still required to pay for the full number. If more people attend than your committed number, they will be served at a cost of $$12$ each. For example, if you commit for $200$ people and $200$ or fewer show up$,$ you still have to pay $$1,400.$ If more than $200$ people attend, you pay $$1,400$ plus $$12$ for each dinner in excess of $200.$ Your judgment about the number attending can be described by a uniform probability distribution with a maximum of $400$ people and a minimum of $100.$

a$)$ Suppose the cost of the banquet is being borne entirely by the Alumni Association, and your objective is to minimize the cost, given that all who attend will be served. How big a prior commitment for dinners should you make to the hotel $($i$.$e$.,$ what is the order quantity$)?$

$[$Round your response up to the nearest integer value, e$.$g$.2\mathrm{.}2$ and $2\mathrm{.}6$ should be both rounded up to $3.]$