Question

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. Simplify completely.

Research Provided

The company's research department has provided the following estimates.

• A demand of 230 banquet attendees can be expected at a dinner plate price of $80.00 each. A demand of 370 banquet attendees can be expected at a dinner plate price of $45.00 each.

• Catering Service A has a fixed cost of $1,900 and a marginal cost of $30 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 $40. All profits will go toward a charity of the committee's choosing.

• Additional spontaneous donations to the charity will be accepted the night of the banquet. Studies estimate that 5% will give $5, 23% will give $20, 18% will give $50, 7% will give $100, and 2% will give $500.

Analysis

Each team should perform the following analyses:

1. Assume the price-demand function is linear. Use the research 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. Sketch a graph on the work page.

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. Sketch a graph on the work page.

4. Determine the break-even quantities for each 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 $112.50, which catering service should the committee recommend in order to earn the most profit for the charity?

7. Decide which catering service your company should choose if the projections yield 200 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 400 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.) Which charity does the committee recommend for receipt of banquet 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 spontaneous donations)?

Answer 1