Answer Happy

A museum has hired you as a consultant to try to help them increase their revenue. They currently charge $65 per year for memberships, and they have 3270 members. They know that every year, most members renew, and that they get some new memberships, as well. The museum director is trying to decide if she should raise or lower prices to increase the revenue as much as possible. The museum conducted a survey and estimates that the percentage of current members who will renew is as given in the table below. ($50 is the minimum that the museum would charge and $120 is the maximum.) Membership $50 $60 $70 $80 $90 $100 $110 $120 fee Percentage 95.1% 93.2% 85.1% 75.7% 64.7% 54% 49.8% 47.5% who renew 1. a) Create a table with the first column being the membership fee, and the second column being the number of people who renew. (Throughout this project, you cannot have fractional people.) b) In addition, the museum estimates from past history that the number of new members that will join at the different prices are as follows:

Membership $50 $60 $70 $80 $90 $100 $110 $120 fee Number of 468 439 391 336 279 214 181 153 new members Create a table with the first column being the membership fee and the second column being the total number of members (new and returning). 2a) Create another table with the first column being the membership fee and the second column being the total revenue using just the data from question 1. b) Determine the maximum revenue from the table. c) How many members would you have? 3. a) Use the table in question 1b and linear regression to find the demand function for membership. Be sure to define your variables. You will use this designation throughout the rest of the project. b) Find the revenue function for the membership using this demand function. c) Use calculus and algebra to determine the membership fee that would maximize your revenue. d) What is the maximum revenue using this method? e) How many members would you have? 4. a) Use the table in question 1b and quadratic regression to find the demand function for membership b) Find the revenue function for the membership. c) Use calculus and algebra to determine the membership fee that would maximize your revenue, d) What is the maximum revenue using this method? e) How many members would you have? 5. a) Use the table in question 1b and cubic regression to find the demand function for membership b) Find the revenue function for the membership c) Use the graph of your function to determine the membership fee that would maximize your revenue. Be sure to explain how you are determining the maximum.

d) What is the maximum revenue using this method? e) How many members would you have? 6. a) Use the table in question 10 and exponential regression to find the demand function for membership. b) Find the revenue function for the membership. c) Use the graph of the function to determine the membership fee that would maximize your revenue. Be sure to explain how you are determining the maximum. d) What is the maximum revenue using this method? e) How many members would you have? 7. You have now computed the maximum revenue five different ways. Which model is best for the museum? You must include the following in your decision: • The fit of the regression equation for that model (if you chose from 3 - 6). compared to the others • How the chosen revenue compares to the current revenue • How your chosen model would work for the museum in both the short-term and long-term Be sure to answer this as though you are the consultant for the museum, so explanations should be informative, but written at a non-calculus level.