$1.$ Sofia$$s Silk Screening produces specialty t$-$shirts that are primarily sold at special events. Sofia is trying to decide how many to produce for an upcoming event. During the event, Sofia can sell t$-$shirts for $$20$ each. However, when the event ends any unsold t$-$shirts are sold for $$4$ each. It costs Sofia$$s $$8$ to make a specialty t$-$shirt. The company$$s estimate of demand is the following:

$$

DEMAND

PROBABILITY

$300$

$0\mathrm{.}05$

$400$

$0\mathrm{.}10$

$500$

$0\mathrm{.}40$

$600$

$0\mathrm{.}30$

$700$

$0\mathrm{.}10$

$800$

$0\mathrm{.}05$

$$

$$

$$

a$)$ What is the service rate $($or optimal fractile$)?$

Note: Round your answer to $2$ decimal places

Service rate$=$

$$

$$

$$

$$

$$

b$)$ How many t$-$shirts should Sofia produce for the upcoming event?

Number of t$-$shirts$=$

$$