Product mix, special order. $($N$.$ Melumad, adapted$)$ Pendleton Engineering makes cutting tools for metalworking operations. It makes two types of tools: R$3,$ a regular cutting tool, and HP$6,$ a high$-$precision cutting tool. R$3$ is manufactured on a regular machine, but HP$6$ must be manufactured on both the regular machine and a high$-$precision machine. The following information is available.

Selling price

Variable manufacturing cost per unit

Variable marketing cost per unit

Budgeted total fixed overhead costs

Hours required to produce one unit on the regular machine

$

$100$

$

$60$

$

$15$

$$350,000$

$1\mathrm{.}0$

HP$6$

$

$150$

$

$100$

$

$35$

$$559,000$

$0\mathrm{.}5$

Additional information includes the following:

$$Pendleton faces a capacity constraint on the regular machine of $50,000$ hours per year.

$$The capacity of the high$-$precision machine is not a constraint.

$$

$$Of the $$550,000$ budgeted fixed overhead costs of HP$6,$ $$300,000$ are lease payments for the high$-$precision machine. This cost is charged entirely to HP$6$ because Pendleton uses the machine exclusively to produce HP$6.$ The lease agreement for the high$-$precision machine can be canceled at any time without penalties.

$$All other overhead costs are fixed and cannot be changed.

$$What product mix$-$that is$,$ how many units of R$3$ and HP$6-$will maximize Pendleton's operating income? Show your calculations.

$$Suppose Pendleton can increase the annual capacity of its regular machines by $15,000$ machine hours at a cost of $$150,000.$ Should Pendleton increase the capacity of the regular machines by $15,000$ machine$-$hours? By how much will Pendleton's operating income increase? Show your calculations.

$$Suppose that the capacity of the regular machines has been increased to $65,000$ hours. Pendleton has been approached by Carter Corporation to supply $20,000$ units of another cutting tool, S$3,$ for $$120$ per unit. Pendleton must either accept the order for all $20,000$ units or reject it totally. S$3$ is exactly like R$3$ except that its variable manufacturing cost is $$70$ per unit. $($It takes one hour to produce one unit of S$3$ on the regular machine, and variable marketing cost equals $$15$ per unit.$)$ What product mix should Pendleton choose to maximize operating income? Show your calculations.