$8.$ Case $-$ Daily Newspaper

A daily newspaper is stocked by a coffee shop so its patrons can purchase and read it while they drink coffee. The newspaper costs $$1\mathrm{.}13$ per unit and sells for $$1\mathrm{.}75$ per unit. If units are unsold at the end of the day, the supplier takes them back at a rebate of $$1$ per unit. Assume that daily demand is approximately normally distributed with $=150$ and $=30.$

Question

$($a$)$ What is your recommended daily order quantity for the coffee shop? $($Round your answer to the nearest integer.$)$

$($b$)$ What is the probability that the coffee shop will sell all the units it orders? $($Round your answer to four decimal places.$)$

$($c$)$ In problems such as these, why would the supplier offer a rebate as high as $$1?$ For example, why not offer a nominal rebate? Find the recommended order quantity at $25$ per unit. $($Round your answer to the nearest integer.$)$