A beverage cans manufacturer makes $3$ types of soft drink cans needed for the beverage producers to fill soft drinks of three different volumes. The maximum availability of the machines' time allotted per day is $90$ hours and the supply of metal is limited to $120$ kg per day. The following table provides the details of the input needed to manufacture one batch of cans.

The manufacturer is interested in maximizing the total profit per day. The sensitivity report provided below was generated using the Excel Solver to determine the optimal solution to this problem. Using the sensitivity report, answer the following questions:

$($a$)$ Optimal production quantity of medium can per day:

$($b$)$ Optimal profit per day:

$($c$)$ Will the current optimal solution stay optimal if the unit profit of large can increases to $60?($Yes or No$)$

$($d$)$ Will the current optimal solution stay optimal if the unit profit of small can decreases to $40?($Yes or No$)$

$($e$)$ Will the current optimal solution stay optimal if we add a constraint that says production quantity of large can per day should be at least $5?($Yes or No$)$

$($f$)$ Will the shadow price of the constraint related to the supply of metal remain valid if we increase the available supply to $130$ kg per day? $($Yes or No$)$

$($g$)$ Calculate the effect on the total profit of decreasing the supply of metal to $115$ kg per day.

$($h$)$ Calculate the effect on the total profit of increasing the machine available time to $95$ hours per day.