A convenience store sells spicy black bean burritos during the weekday lunch hour. It charges $$6\mathrm{.}00$ for each burrito and the burritos

are purchased from a local manufacturer for $$1\mathrm{.}75$ each. Any burritos left at the end of the day are disposed of and have no salvage

value. The manufacturer's production cost is $$1\mathrm{.}00$ per burrito. Burrito demand follows a normal distribution with a mean of $200$ and a

standard deviation of $40.$

Suppose burrito customers buy their snack somewhere else if the convenience

a$.$ store is out of stock. How many burritos should the convenience store order for the

lunch crowd?

b$.$ What quantity of burritos should be ordered to maximize the expected supply chain

profit?

$($Round your answer to the nearest unit.$)$

What is the minimum buyback price the manufacturer could offer the conveneince

c$.$ store in order to get them to order the optimal supply chain quantity of burritos from

part $($b$)?$

$($Round your answer to two decimal places.$)$

d$.$ Suppose the Kiosk makes $300$ burritos. What is the probability that some customer

is unable to purchase a burrito?

$($Round your answer to $4$ decimal places.$)$

e$.$

If the Kiosk wants to be sure it has inventory for its customers with at least a $0\mathrm{.}985$

probability, how many burritos should it make?

$($Round your answer to the nearest unit.$)$

Suppose that any customer unable to purchase a burrito settles for a lunch of Pop$-$

f$.$

Tarts. Pop$-$Tarts sell for $75$ and cost the Kiosk $25.($As Pop$-$Tarts are easily

stored, the Kiosk never runs out.$)$ Assuming that the Kiosk management is

interested in maximizing profits, how many burritos should it make?

$($Round your answer to the nearest unit.$)$