The Company you are working for has decided to put on a banquet to raise money for a charity and must decide between two catering services to supply the food and servers. You have been assigned to the Banquet Planning Committee. You should work in a group with up to 4 members, cach bringing their own individual strengths to the group. You will submit your work, neatly hand-written and clearly labeled. You must use algebra to solve the problems (not graphing calculator features). To best maintain accuracy, round only your final answers. Simplify completely. Research Provided: The Company's rescarch department has provided the following estimates: A demand of 285 banquet attendees can be expected at a dinner plate price of $71.00 cach. A demand of 387 banquet attendees can be expected at a dinner plate price of $39.00 each. Catering Service A has a fixed cost of $2,300 and a marginal cost of S31 for each plate. Catering Service B has a fixed cost of $3,000 and a marginal cost of $22 for each plate. Costs for both caterers include the food, drinks, plates, utensils, tablecloths, glasses, crew, and cleanup. Dinner plates will only be sold as an entire unit. To justify company resources and to ensure the event will benefit the charity, the CEO insists the tickets be sold for no less than S30. All profits will go toward a charity of the committee's choosing. + Additional spontancous donation to the charity will be accepted the night of the banquet. Studies estimate that 7% will give S5, 19% will give S18, 11% will give $33, 2% will give S130, and 1% will give $500. Analysis: Your group should perform the following analyses: 1. Assume the price-demand function is linear. Use the rescarch estimates to find the relationship between the price p, and the number of banquet attendees demanded, x. Find the relevant domain. 2. Find the revenue function, R(x), in terms of the number of banquet attendees x. Find the relevant domain by considering realistic limitations on the number of attendees and on price. Provide a sketch of graph. 3. Assume the cost function is linear and use the research estimates to find the cost function for each of the two possible catering services in terms of the number of banquet attendees x. Provide a sketch of graph. 4. Determine the break-even quantities for cach of the two possible catering services. 5. Find the Profit function, P(x), for each of the two possible catering services in terms of the number of banquet attendees x. 6. If it is projected that there will be 100 tickets sold at a dinner price of S110, which catering service should the committee recommend in order to eam the most profit for the charity? catering your com if the projections yield 260 attendees. Include the ticket price at this demand. 8. Find the average cost function for Catering Service A. Evaluate the average cost per attendee if 50 tickets are purchased. Evaluate the average cost per attendee if 600 tickets are purchased. 9. On average, how much can you expect to receive in spontaneous donations for each banquet attendee? (Determine the per person expected value (weighted average) of donations.) 10. Overall recommendations: Which catering service does your committee recommend in order to obtain the most profit for the charity? How many people should attend the banquet to earn that profit and what profit can be expected from the banquet ticket sales? What price would you recommend for each dinner plate (ticket)? What dollar amount is expected from spontaneous donations? What will be the expected total amount raised for the charity (including the dinner ticket profits and spontancous donations)? *Here is the technical worksheet: Show all solutions at the last page- take a screenshot of your solutions then paste at the last page of the worksheet. Technical Worksheet 1. The linear price-demand function is p = D(x) = Domain: 2. The revenue function is R(x) - Domain: 3. The cost functions for each of the two possible catering services in terms of x are CA (x) - CB (x) = and 4. Catering Service A will at least break even at a minimum of banquet attendees and a maximum of banquet attendees. Catering Service B will at least break even at a minimum of banquet attendees and a maximum of banquet attendees. 5. The profit functions for each of the possible Contractors in terms of x are PA (x) PB (x) = and maximize the banquet's profit since the profit using Catering Service A is $ $ and the profit using Catering Service B is 7. If the projected sales are 260 banquet tickets at a unit price of your company should choose Catering Service in order to maximize the banquet's profit since the profit using Catering Service A is $ Service B is $ and the profit using Catering 8. The average cost function for Catering Service A is CA(x)= a. The average cost per attendee if 50 tickets are purchased is CA(50) $ b. The average cost per attendee if 600 tickets are purchased is CA(650) $ 9. On average, the expected value of spontaneous donations is $ per attendee. 10. The Committee would like to recommend Catering Service Number of Attendees for Maximum Profit: Maximum Charity Ticket Profit: S Optimal Number of Attendees: Optimal Charity Ticket Profit: $ Banquet Ticket Price for Maximum Profit: $ Optimal Banquet Ticket Price: $ Maximum Expected Spontaneous Donations: S Optimal Spontaneous Donations: $