Sub Place, a small sandwich store, is located on a busy corner near many other businesses. The shop's busiest time is during the mid$-$day period. Rent for the location is $$1000$ per month, and wages amount to $$2000$ per month. Variable costs consist of supplies and sandwich ingredients that cost $$2\mathrm{.}14$ per sub sandwich. The subs are to be sold at a price of $$5\mathrm{.}69$ each. Answer each of the following independent questions.

$($a$)$ How many sandwiches must the shop sell to break even?

$($b$)$ If the shop increases variable costs by $$0\mathrm{.}23$ per sandwich, how many sandwiches must it sell to break even?

$($c$)$ If the rent increases by $12\%,$ what would the profit be if the shop sold $1575$ units?

$($d$)$ If the sandwich price was reduced by $$0\mathrm{.}23,$ how many sandwiches must the shop sell to make $$1082$ profit?

$($a$)$ The shop must sell $846$ sandwiches to break even.

$($Round up to the nearest whole number.$)$

$($b$)$ If variable costs increase by $$0\mathrm{.}23$ per sandwich, the number of sandwiches required to break even is $904.$

$($Round up to the nearest whole number.$)$

$($c$)$ If the rent increased by $12\%$ and the shop sold $1575$ units, the shop would make a$($n$)!$ profit.

$($Round to the nearest cent as needed.$)$