Mack's Juices produces and bottles a line of fruit juices. The manufacturing process entails mixing and adding juices and other ingredients at the bottling plant, which is a part of Blending Division. The finished product is packaged in a company$-$produced glass bottle and packed in cases of $24$ bottles each. Because the appearance of the bottle heavily influences sales volume, Mack's developed a unique bottle production process at the company$$s container plant, which is a part of Packaging Division. Blending Division uses all of the container plant$$s production. Each division $($Blending and Packaging$)$ is considered a separate profit center and evaluated as such. As the new corporate controller, you are responsible for determining the proper transfer price to use for the bottles produced for Blending Division. At your request, Packaging Division$$s general manager asked other bottle manufacturers to quote a price for the number and sizes demanded by Blending Division. These competitive prices follow: Volume Total Price Price per Case $100,000$ equivalent casesa $ $2,610,000$ $ $26\mathrm{.}10200,0004,650,00023\mathrm{.}25300,0006,210,00020\mathrm{.}70$ a An equivalent case represents $24$ bottles. Packaging Division$$s cost analysis indicates that it can produce bottles at these costs. Volume Total Cost Cost per Case $100,000$ equivalent cases $ $2,325,000$ $ $23\mathrm{.}25200,0003,975,00019\mathrm{.}88300,0005,625,00018\mathrm{.}75$ These costs include fixed costs of $$675,000$ and variable costs of $$16\mathrm{.}50$ per equivalent case. These data have caused considerable corporate discussion as to the proper price to use in the transfer of bottles from Packaging Division to Blending Division. This interest is heightened because a significant portion of a division manager$$s income is an incentive bonus based on profit center results. Blending Division has the following costs in addition to the bottle costs: Volume Total Cost Cost per Case $100,000$ equivalent cases $ $1,800,000$ $ $18\mathrm{.}00200,0002,400,00012\mathrm{.}00300,0003,900,00013\mathrm{.}00$ The corporate marketing group has furnished the following price$$demand relationship for the finished product: Sales Volume Total Sales Revenue Sales Price per Case $100,000\mathrm{.}00$ equivalent cases $ $6,450,000$ $ $64\mathrm{.}50200,000\mathrm{.}0011,700,00058\mathrm{.}50300,000\mathrm{.}0014,850,00049\mathrm{.}50$ Required: a$.$ Mack's Juices has used market price$$based transfer prices in the past. Using the current market prices and costs and assuming a volume of $300,000$ cases, calculate operating profits for the Packaging Division, Blending Division, and Mack's Juices. b$-1.$ Calculate operating profits for Packaging, Blending, and Mack's Juices for volumes of $100,000,200,000$ and $300,000$ cases. b$-2.$ Which volume of production is the most profitable for Packaging, Blending, and Mack's Juices?