Farrugia Corporation produces two intermediate products, A and B$,$ from a common input. Intermediate product A can be further processed into Product $x.$ Intermediate product B can be further processed into Product $Y.$ The common input is purchased in batches that cost $$45$ each and the cost of processing a batch to produce intermediate products A and $B$ is $$18.$ Intermediate product A can be sold as is for $$28$ or processed further for $$17$ to make Product $x$ that is sold for $$41.$ Intermediate product B can be sold as is for $$61$ or processed further for $$34$ to make Product $Y$ that is sold for $$82.$

Required:

a$.$ Assuming that no other costs are involved in processing the common input or in selling products, what is the profit $($loss$)$ from processing one batch of the common input into the products $x$ and $Y?$

b$.$ What is the Financial advantage $($disadvantage$)$ from further processing? Should each of the intermediate products, A and B$,$ be sold as is or processed further?