Math 1030-007 Fall 2010 Extra Credit Project Expectations for Extra Credit Project Summary: In this project you will have the opportunity to apply what you have learned in Calculus to answer questions about real world problems. You will be expected to use the principles of calculus to solve the problems mathematically, and you will also be expected to do enough background research to collect the information necessary to solve the problems. In many cases, you may even need to learn about a new topic in Calculus that we did not cover in this course but is very closely related to topics that we have covered. The project is a report on an investigation of a question. This means your report should begin with an introductory paragraph in which you discuss the question or problem that you are examining and introduce briefly how you approached your work, perhaps give a brief statement of something that you discovered or a conclusion you will examine in more detail later in the report. The next part of your report will be the main body in which you discuss your work, give your analysis, and demonstrate your results. The last part of your report should be a nice summary of your findings and conclusions backed up by the evidence you presented in the middle part of your report. Instructions: - You may work on this project individually or in a group of no more than three people. - Select one of the approved topics for your project. - Projects must be typewritten on 81/2 x 11 paper, double spaced, 12 point font. - Math formulas, equations, diagrams can be written in by hand only if done neatly. - Equations and calculations should be accompanied by explanations given in complete sentences of what these calculations mean. - If tables or graphs are attached at the back, then the main body of the report should include correct references to page numbers, figures, and appendices. - Sources used for information or data should be described and adequate references for these sources given. Material taken from references must be summarized in one's own words. - It is recommended that you edit your report at least once before submitting it for grading. Things to look for in the editing process include typographical errors, grammar and spelling errors, and awkward writing. If you are working in a group, all members of the group should read the final draft of the project, approve it before it is submitted, and sign their name on the title page. If there is group member who has made no contribution to the project, the instructor should be informed and that student's name not included in the report. Due Date: Monday, December 6, 2010 Grading: - Mechanics: 20% (spelling, grammar, punctuation, citations, formatting, etc.) - Mathematics & Analysis: 80% (accurate calculations, correct use of concepts, proper reasoning, logical flow of ideas etc.)Math 1030-007 Fall 2010 Extra Credit Project Option #1: (Startup Company) In this course you learned that one of the most important applications of the derivative is in optimization problems (see section 3.4). We went over many cool examples in class but in this project you will have the opportunity to do a much more in depth exploration real world optimization problems. You may need to do some research in order to determine some of the formulas and numbers you will need in order to answer the following questions. Be sure to cite your sources. (a) You wish to open a small business in Salt Lake City where you will offer tutoring services for high school mathematics. You have done some market analysis and have found that the demand function is where p is the price you charge for one hour of tutoring and D(p) is the combined number of tutoring hours that all of your customers want at the give price each month. In this situation, what should you set your price at in order to maximize your profit (ignore operating costs for the time being). (b) You have several tutors working under you, and the number of hours they are willing to work each month depends on how much you pay them. Suppose your supply-side analysis shows that the supply function for tutors is where q is the price you pay each tutor per hour and S(q) is the combined number of hours each tutor is willing to work at that price each month. Using your result from part (a), determine the price to pay your tutors that will maximize profit for your company. Use your common sense to make certain simplifying assumptions if necessary. (c) You decide it would be a good idea to try and advertise your company to the public. Do some research to figure out how much it would cost per month to run an ad on the local television channel and to get a large ad in the local yellow pages. You do some market analysis and determine that the function for returns for advertising is where p is the price of tutoring per hour that you advertise and A(p) is the number of additional hours demanded by customers each month as a result of your advertisements. What is your revenue function in terms of price when you place this ad on TV? What is your profit function, ignoring for the moment the cost of paying your tutors. What price should you set to maximize your profit? At this optimal price, what is the price that you should pay your tutors? Do you make more money with or without advertising in this case? (d) Do some research to figure out how much it would cost each month to rent 1,500 square feet of office space in Salt Lake City, include in this calculation the costs for utilities, maintenance, insurance, and relevant licenses. Use your imagination to determine what sorts of overhead costs you might have just for getting the company started, as well as operating costs and variable costs. Do the necessary research to reasonably well approximate how much each of these things would cost. Include all of this in your cost function and use this information to write down your profit function. Then determine the price you would need to set to maximize your profit, and use that to determine the price you would need to pay your tutors in order to maximize profit. Is it worthwhile to start up this company? What would be your annual salary if you kept 70% of the profits and invested the