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 $125$ hours, and the grinder is available for $115$ hours. No more than $200$ units of component $3$ can be sold, but up to $800$ units of each of the other components 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$ $$6$

$2$ $$18$ $$6$ $$4$

$3$ $$27$ $$16$ $$5$

$($a$)$

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

Formulate and solve for the recommended production quantities that will maximize contribution to profit.

$($b$)$

What are the objective coefficient ranges $($in dollars$)$ for the three components?

Interpret these ranges for company management.

$($c$)$

What are the right$-$hand$-$side ranges?

Interpret these ranges for company management.

$($d$)$

If more time could be made available on the grinder, how much would it be worth?

$($e$)$