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A small firm intends to increase the capacity of a bottleneck operation by adding a new machine. Two alternatives, A and B$,$ have been identified, and the associated costs and revenues have been estimated. Annual fixed costs would be $$80,000$ for A and $$50,000$ for B; variable costs per unit would be $$8$ for A and $$10$ for B; and revenue per unit would be $$13.$

$1.$ Determine each alternative$$s break$-$even point in units.

$2.$ At what volume of output would the two alternatives yield the same profit?

$3.$ If expected annual demand is $20,000$ units, which alternative would yield the higher profit?