Cantel Company produces cleaning compounds for both commercial and household customers. Some of these products are produced as part of a joint manufacturing process. For example, GR$37,$ a coarse cleaning powder meant for commercial sale, costs $$2\mathrm{.}70$ a pound to make and sells for $$2\mathrm{.}80$ per pound. A portion of the annual production of GR$37$ is retained for further processing in a separate department where it is combined with several other ingredients to form SilPol, which is sold as a silver polish, at $$5\mathrm{.}00$ per unit. The additional processing requires $1/4$ pound of GR$37$ per unit; additional processing costs amount to $$3\mathrm{.}20$ per unit of SilPol produced. Variable selling costs for SilPol average $$0\mathrm{.}40$ per unit. If production of SilPol were discontinued, $$6,300$ of costs in the processing department would be avoided. Cantel has, at this point, unlimited demand for, but limited capacity to produce, product GR$37.$

Required:

$1.$ Calculate the minimum number of units of SilPol that would have to be sold in order to justify further processing of GR$37.$

$2.$ Assume that the cost data reported for GR$37$ are obtained at a level of output equal to $5,700$ pounds, which is the maximum that the company can produce at this time. What is the expected operating income $($loss$)$ under each of the following scenarios: $($a$)$ all available capacity is used to produce GR$37,$ but no SilPol; $($b$)7,000$ units of SilPol are produced, with the balance of capacity devoted to the production and sale of GR$37$; $($c$)9,000$ units of SilPol are produced, with the balance of capacity devoted to the production and sale of GR$37$; and $($d$)11,000$ units of SilPol are produced, with the balance of capacity devoted to the production and sale of GR$37.$