Part $1$: Bagel Shop $(14$ points$)$

Sofia operates a small bagel shop in Irvine, near campus.$$ Batches of bagels are baked fresh every morning.$$ To simplify your analysis, you can assume that Sofia only sells one variety of bagel $($your favorite kind!$).$ Sofia estimates that it costs $$20$ in labor, materials, and energy to produce $24$ bagels $($she always bakes bagels in batches of $24).$ She sells each single bagel for $$1\mathrm{.}95.$ Demand is typically high, as there are many students near campus who like bagels.$$ Daily demand is Normally distributed with a mean of $300$ and a standard deviation of $134.$ Whenever Sofia has unsold bagels leftover at the end of the day, she sells them to the local grocery store for $$0\mathrm{.}50$ each. The bagels sold by the grocery store won't be as fresh, but they won't be labelled as "Sofia$$s bagels", so this won't erode Sofia$$s brand. Sofia needs your help to determine how many bagels she should bake daily.

Question $1.$a:

What is the underage cost and overage cost $($per bagel$)?$

$$

$$

Underage Cost $=$

$$

Overage Cost $=$

$$

$$

$$

Question $1.$b:

Using the underage and overage costs you just computed, compute the critical ratio. Then, calculate how many bagels Sofia should bake daily to maximize her profit.$($Round up to a multiple of $24,$ since Sofia bakes bagels in batches of $24)$

$$

$$

Critical Ratio $=$

$$

$$

$$

# Bagels to Bake Daily $=$

$$

$$

Question $1.$c:$$ If in the morning Sofia bakes $240$ bagels $($i$.$e$.,10$ batches$),$ how many bagels would she expect to sell to customers that day?

Question $1.$d:$$ If in the morning Sofia bakes $240$ bagels $($i$.$e$.,10$ batches$),$ how many bagels would she expect will be left over at the end of the day, to be sold to the grocery store?$$

$$

Question $1.$e:$$ If in the morning Sofia bakes $240$ bagels $($i$.$e$.,10$ batches$),$ in expectation how many customers will arrive at her shop that day to find all of the bagels have already been sold?$($Assume each customer wishes to buy exactly one bagel$)$

$$

Question $1.$f: If in the morning Sofia bakes $240$ bagels $($i$.$e$.,10$ dozen$),$ what is her expected profit?$$

$$

$$

Question $1.$g: Now assume that instead of being Normally distributed, daily demand is Uniformly$-$distributed between $200$ and $400($notice the mean is still $300).$ How many bagels should Sofia bake each day now? $($Round up to a multiple of $24,$ since Sofia bakes bagels in batches of $24).$

$$

$$

Question $1.$h: Now assume we have historical data for daily demand $($all on the same day of the week, so we don't need to worry about seasonal effects$).$ Demands on these ten days were $($sorted in order$)$: $110,156,192,241,264,276,288,312,324,345.$ How many bagels should Sofia bake each day now? $($Round up to a multiple of $24,$ since Sofia bakes bagels in batches of $24).$