Problem $3.$ A gourmet store sells a perishable delicacy fish, which is brought fresh daily from fishermen just prior to the

start of the business day. Unsold fish must be disposed of at the end of the business day. One pound of fish costs the

store $$28\mathrm{.}50$ per pound, and sells for $$150$ per pound. The store sells daily leftover fish to a cat food company at $$20$ per

pound, but there is a transportation cost of $$11\mathrm{.}40$ per pound of unsold inventory. The store estimates that daily demand

follows a continuous uniform distribution between $51$ and $250$ pounds $($inclusive$)$ of fish per day, that is$,$ the probability of

any quantity of fish, $Q,$ between $51$ pounds and $250$ pounds is $f(Q)=\frac{1}{200},$ and the corresponding $cdf$ is $F(Q)=\frac{Q-50}{?}$

$200$ for $Q$ in the same range.

Answer the following questions:

$3\mathrm{.}1)(5$ points$)$ Write down the model name and list all the parameters of the problem and their numerical values.

$3\mathrm{.}2)(10$ points$)$ What is the optimal quantity of fish $($rounded off to the nearest pound, if not integer$)$ the store should

stock each business day?

$3\mathrm{.}3)(10$ points$)$ If the daily demand actually has a normal distribution with mean $150$ and standard deviation $20,$ what is

the optimal quantity of fish $($rounded off to the nearest pound$)$ the store should stock each business day?