Dear Calculus Students:Thank you for your assistance with my paper clip scheduling problem! You cleared up mymisunderstanding, and I have not made any more scheduling mistakes (nor had my pay dockedagain.) My boss, Mr. Tom Tiddlywinks, has been so happy with my performance that he hasgiven me some additional responsibilities, with the hint that a promotion may be in my future.You can imagine how delighted I am. One of these new tasks, however, has challenged mewith some of the mathematics involved. Since your help was so invaluable before, I am writingto request your assistance again.Our forwardthinking company, Tiddlywinks & Sons Manufacturing Corporation, is developingtwo new products that may revolutionize the world of office products. We are getting ready forproduction and will be hiring at least six new workers to start after Thanksgiving. It is myresponsibility to come up with a production plan that will lead to the highest profit possible.I am not allowed to divulge any details about our new products, (everything is top secret as wedont want our competition to know what were up to). Let me call our new items product X andproduct Y, and I need to determine how many units of each product to produce. We expect torealize a profit of a dollars per unit on product X and b dollars per unit with product Y (asdetermined by our marketing department).The actual number of units that we will be able to produce is, however, limited by the number ofhours that the workers have available, which we call personhours. Once weve made our newhires, we will have N total personhours that we can devote to the production of products X andY. (Again, I apologize, but Im not allowed even to tell you what the number N is.) Each unit ofX requires c personhours to produce. For the second product, however, economies of scaleare more pronounced, so that the number of personhours required to produce y units of Y isproportional to y , with a constant of proportionality d. 1/2In order for our company to be successful (and for me to have a chance at a promotion), it isimperative that we determine the best production strategy for these two products. When Ipresent my suggested production plan, I will also need to justify to Mr. Tiddlywinks why this willbe the best possible plan. I have heard through the lunchroom gossip that Mr. Tiddlywinks isexpecting that the highest profit will result if we produce twice as many of product Y as we do ofproduct X (because of the economies of scale mentioned above.) I, however, think the situationwill be much more complicated, and our plan will depend on all the constants that Ivedescribed. But I dont know just how to work it all out, and so Im hoping that you can help me. Idepend on you to explain everything to me clearly. Since we are gearing up for productionsoon, I would appreciate an answer as soon as possible, but certainly no later than November20.Yours sincerely,Fred Fellows