$($Penne Pesto$)$ Penne Pesto is a small restaurant in the financial district of San Francisco. Customers order from a variety of pasta dishes. The restaurant has $50$ seats and is always full during the four hours in the evening. It is not possible to make resexvations at Penne; most guests show up spontaneously on their way home from work. If there is no available seat, guests simply move on to another place.

On average, a guest spends $50$ minutes in the restaurant, which includes $5$ minutes until the guest is seated and the waiter has taken the order, an additional $10$ minutes until the food is served, $30$ minutes to eat, and $5$ minutes to handle the check$-$out $($including waiting for the check, paying, and leaving$).$ It takes the restaurant another $10$ minutes to clean the table and have it be ready for the next guests $($of which there are always plenty$).$ The average guest leaves $$20$ at Penne, including food, drink, and tip $($all tips are collected by the restaurant; employees get a fixed salary$).$

The restaurant has $10$ waiters and $10$ kitchen employees, each earning $$90$ per evening $($including any preparation, the $4$ hours the restaurant is open, and clean$-$up$).$ The average order costs $$5\mathrm{.}50$ in materials, including $$4\mathrm{.}50$ for the food and $$1$ for the average drink. In addition to labor costs, fixed costs for the restaurant include $$500$ per day of rent and $$500$ per day for other overhead costs.

The restaurant is open $365$ days a year and is full to the last seat even on weekends and holidays. There is about $$200,000$ of capital tied up in the restaurant, largely consisting of furniture, decoration, and equipment.

a$.$ How many guests will the restaurant serve in one evening? $[6\mathrm{.}2]$

b$.$ What is the return on invested capital $($ROIC$)$ for the owner of the restaurant? $[6\mathrm{.}2]$

c$.$ Assume that you could improve the productivity of the kitchen employees and free up one person who would be helping to clean up the table. This would reduce the clean$-$up to $5$ minutes instead of $10$ minutes. What would be the new ROIC? $[6\mathrm{.}3]$

d$.$ What would be the new ROIC if overhead charges could be reduced by $$100$ per day? $6\mathrm{.}3$