Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times $($in minutes$)$ required on each machine are as follows:

Machine

Component Shaper Grinder

$164$

$245$

$342$

The shaper is available for $120$ hours, and the grinder for $110$ hours. No more than $200$ units of component $3$ can be sold, but up to $1,000$ units of component $1$ and up to $1,000$ units of component $2$ can be sold. In fact, the company already has orders for $600$ units of component $1$ that must be satisfied. The per unit selling price and per unit variable costs for each of the three components are as follows:

Component Selling price Material Cost Labor Cost

$1$ $$25$ $$12$ $$5$

$2$ $$18$ $$8$ $$4$

$3$ $$27$ $$13$ $$5$

$($a$)$ For each component, calculate the profit margin $($profit margin $=$ selling price $-$ material cost $-$ labor cost$).$